Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
ANSWER KEY<br />
<strong>Preparing</strong> <strong>for</strong> <strong>the</strong><br />
REGENTS EXAMINATION<br />
GEOMETRY<br />
AMSCO<br />
AMSCO SCHOOL PUBLICATIONS, INC.<br />
315 Hudson Street, New York, N.Y. 10013<br />
N 81 CD
Compositor: Monotype LLC<br />
Please visit our Web site at:<br />
www.amscopub.com<br />
Copyright © 2008 by Amsco School Publications, Inc.<br />
No part of this book may be reproduced in any <strong>for</strong>m without<br />
written permission from <strong>the</strong> publisher.<br />
Printed in <strong>the</strong> United States of America<br />
1 2 3 4 5 6 7 8 9 10 12 11 10 09 08 07
Contents<br />
Chapter 1: Essentials of <strong>Geometry</strong> 1<br />
Chapter 2: Logic 3<br />
Chapter 3: Introduction to Geometric Proof 8<br />
Chapter 4: Congruence of Lines, Angles, and Triangles 12<br />
Chapter 5: Congruence Based on Triangles 18<br />
Chapter 6: Trans<strong>for</strong>mations and <strong>the</strong> Coordinate Plane 28<br />
Chapter 7: Polygon Sides and Angles 34<br />
Chapter 8: Slopes and Equations of Lines 39<br />
Chapter 9: Parallel Lines 44<br />
Chapter 10: Quadrilaterals 56<br />
Chapter 11: <strong>Geometry</strong> of Three Dimensions 68<br />
Chapter 12: Ratios, Proportion, and Similarity 72<br />
Chapter 13: <strong>Geometry</strong> of <strong>the</strong> Circle 79<br />
Chapter 14: Locus and Constructions 87<br />
Cumulative Reviews 92<br />
Practice <strong>Geometry</strong> <strong>Regents</strong> <strong>Examination</strong>s 138
Essentials of <strong>Geometry</strong><br />
1-1 Undefined Terms<br />
(page 2)<br />
1 (1) finite<br />
2 (2) infinite<br />
3 (3) empty<br />
4 A B<br />
5 A given line lies on an infinite number of<br />
planes.<br />
6 a and b<br />
m<br />
1-2 Real Numbers and<br />
Their Properties<br />
(pages 4–5)<br />
1 (4) 8<br />
2 (4) 5<br />
3 (3) 0n 0<br />
4 (1) a(b c) ab ac<br />
5 (1) additive inverse<br />
6 Commutative<br />
7 Multiplicative identity<br />
8 Commutative<br />
9 Additive identity<br />
10 Associative<br />
11 Associative<br />
<br />
p<br />
CHAPTER<br />
1<br />
12 Commutative<br />
13 Distributive<br />
14 Additive inverse<br />
15 Distributive<br />
16 Multiplicative inverse<br />
17 1<br />
18 0<br />
19 0<br />
20 Division by zero is undefined.<br />
1-3 Lines and Line<br />
Segments<br />
(pages 6–7)<br />
1 8<br />
2 4<br />
3 −−<br />
BD and −−<br />
CE<br />
4 They must all lie on <strong>the</strong> same line.<br />
5 They <strong>for</strong>m a triangle.<br />
6 Point S is <strong>the</strong> midpoint. It is halfway between<br />
P and T.<br />
7 Correct. The length of −−<br />
AB is equal to <strong>the</strong><br />
length of −−−<br />
CD .<br />
8 Incorrect notation<br />
9 Incorrect notation<br />
10 Correct. −−<br />
AB is congruent to −−−<br />
CD .<br />
11 −−−<br />
CPD bisects −−−<br />
APB . It divides it into two segments<br />
with <strong>the</strong> same measure, 7.<br />
12 MS 6<br />
1-3 Lines and Line Segments 1
1-4 Angles<br />
(pages 9–10)<br />
1 (4) It is a portion of a line, beginning with<br />
point M.<br />
2 (1) DAE<br />
3 (4) XYZ is a 180 angle.<br />
4 (3) equal to 90<br />
5 (2) three acute angles<br />
6 Acute angles: JKM, MKN, NKO, OKL,<br />
MKO<br />
Obtuse angles: JKO, MKL<br />
Right angles: JKN, NKL<br />
Straight angle: JKL<br />
7 mLKO 35; mMKN 30; mMKO 85<br />
8<br />
D<br />
9<br />
10<br />
11<br />
E F<br />
R<br />
V<br />
H<br />
B<br />
12 a and b<br />
E<br />
A<br />
G<br />
B C<br />
I<br />
2 Chapter 1: Essentials of <strong>Geometry</strong><br />
I<br />
T<br />
1-5 Preliminary Angle and<br />
Line Relationships<br />
(pages 11–12)<br />
1 a mDBC 90<br />
b DBC ABD<br />
2 a mONP 45<br />
b ONP MNO<br />
3 mCBD<br />
4 mCBD<br />
5 mABD mABC mCBD<br />
90 45<br />
135<br />
6 l n<br />
7 x 8<br />
8 y 9.5<br />
9 mCAT 28<br />
10 mAFC 46<br />
1-6 Triangles<br />
(page 16)<br />
1 False<br />
2 True<br />
3 False<br />
4 False<br />
5 True<br />
6 The length of each leg of <strong>the</strong> isosceles<br />
triangle is 18.<br />
7 a The length of each side of <strong>the</strong> triangle<br />
is 16.<br />
b y 7<br />
8 mA 40; mB 70; mC 70<br />
9 Check students’ drawings of isosceles<br />
triangle ABC with vertex at C.<br />
10 Check students’ drawings of right triangle<br />
PQR with hypotenuse ___<br />
PQ .<br />
11 Check students’ drawings of acute triangle<br />
FGH that is scalene.<br />
12 −−−<br />
CD and −−<br />
CE<br />
Chapter Review (pages 16–17)<br />
1 Point<br />
2 Line, length<br />
3 Plane, length and width<br />
4 9<br />
5 Only <strong>the</strong> square roots of perfect square numbers<br />
are rational.
6 a −−<br />
XZ −−<br />
YZ<br />
b ZXY and ZYX<br />
7 Coordinate of B is 2.<br />
8 mKJL 36<br />
9 G is <strong>the</strong> midpoint of −−<br />
FH .<br />
10 Bisectors pass through <strong>the</strong> same point, <strong>the</strong><br />
midpoint of <strong>the</strong> line segment. Distinct parallel<br />
lines have no points in common.<br />
CHAPTER<br />
2<br />
2-1 Statements, Truth<br />
Values, and Negations<br />
(pages 20–21)<br />
1 (2) a false closed sentence<br />
2 (2) a false closed sentence<br />
3 (4) not a statement<br />
4 (1) a true closed sentence<br />
5 (4) 4 3 is not greater than 2.<br />
6 (2) The coat is not blue.<br />
7 (2) false<br />
8 (1) 3 6 7<br />
9 Any open sentence<br />
10 A triangle has three sides.<br />
11 Statistics is <strong>the</strong> study of analyzing data.<br />
12 Algebra is <strong>the</strong> study of operations on sets of<br />
numbers.<br />
13 Trigonometry is <strong>the</strong> study of triangles.<br />
14 <strong>Geometry</strong> is <strong>the</strong> study of shapes and sizes.<br />
15 “Chicago is not a city in Indiana.”<br />
Original statement is false.<br />
Negated statement is true.<br />
16 “The sun does not rise in <strong>the</strong> west.”<br />
Original statement is false.<br />
Negated statement is true.<br />
Logic<br />
11 Straight angles: BCE and ACD<br />
12 DEF<br />
13 Right triangles: ABC and CED<br />
14 CFE<br />
15 −−<br />
BD , −−<br />
FD , −−<br />
FE , and −−<br />
BE are choices.<br />
16 −−<br />
BD<br />
17 “January is not a winter month.”<br />
Original statement is true.<br />
Negated statement is false.<br />
18 “The area of a rectangle is not length times<br />
width.”<br />
Original statement is true.<br />
Negated statement is false.<br />
19 “Each state does not have two senators.”<br />
Original statement is true.<br />
Negated statement is false.<br />
20 “It is not <strong>the</strong> case that all real numbers are<br />
rational.”<br />
Original statement is false.<br />
Negated statement is true.<br />
2-2 Compound Statements<br />
(pages 23–24)<br />
1 (3) 3 7 10 or 4 7 3<br />
2 (1) Albany is not <strong>the</strong> capital of New York<br />
and is located on Long Island.<br />
3 (2) 5 is not an odd number or 6 4 12.<br />
4 (4) Daylight saving time ends in November<br />
or <strong>the</strong> clock is not turned back one hour.<br />
5 False<br />
2-2 Compound Statments 3
6 True<br />
7 True<br />
8 True<br />
9 True<br />
10 Answers will vary. One possible answer is:<br />
Every square has four sides or every triangle<br />
has three sides. True<br />
11 Answers will vary. One possible answer is:<br />
The sum of <strong>the</strong> measures of <strong>the</strong> angles of<br />
every triangle is 180 and every triangle has<br />
four angles. False<br />
2-3 Conditionals<br />
(pages 26–27)<br />
1 (2) You take this medicine.<br />
2 (1) I will stay in this afternoon.<br />
3 (1) true<br />
4 (1) true<br />
5 (1) true<br />
6 Conditional<br />
7 Conjunction<br />
8 Conditional<br />
9 Disjunction<br />
10 Negation<br />
11 Answers will vary. x 18 is one possible<br />
answer.<br />
12 x 2<br />
13 Any positive number or zero<br />
14 Any perfect square that is not even<br />
15 Answers will vary. x √ 2 is one possible<br />
answer.<br />
16 True<br />
17 False<br />
18 The sun is<br />
shining.<br />
It is not<br />
raining.<br />
If <strong>the</strong> sun is<br />
shining, it is<br />
not raining.<br />
Sunny T T T<br />
Sun<br />
showers<br />
T F F<br />
Cloudy F T T<br />
Stormy F F T<br />
4 Chapter 2: Logic<br />
2-4 Converses, Inverses,<br />
and Contrapositives<br />
(pages 28–29)<br />
1 (1) If you stop your car, <strong>the</strong>n <strong>the</strong> traffic light<br />
is red.<br />
2 (3) If <strong>the</strong> sum of two numbers is not even,<br />
<strong>the</strong>n <strong>the</strong> numbers are not both even.<br />
3 (1) If <strong>the</strong> merrygoround was not oiled, <strong>the</strong>n<br />
it will squeak.<br />
4 (2) If we are on vacation, it is summer.<br />
5 (4) If a triangle is not equilateral, <strong>the</strong>n it<br />
does not have three congruent angles.<br />
6 (2) If I didn’t turn on <strong>the</strong> air conditioner,<br />
<strong>the</strong>n <strong>the</strong> house isn’t cool.<br />
7 (2) disjunction, p ∨ q<br />
8 Converse: If school is open, <strong>the</strong>n it is September.<br />
Inverse: If it is not September, <strong>the</strong>n school is<br />
not open.<br />
Contrapositive: If school is not open, <strong>the</strong>n it is<br />
not September.<br />
9 Converse: If <strong>the</strong> figure has four sides, <strong>the</strong>n it<br />
is a square.<br />
Inverse: If <strong>the</strong> figure is not a square, <strong>the</strong>n it<br />
does not have four sides.<br />
Contrapositive: If <strong>the</strong> figure does not have<br />
four sides, <strong>the</strong>n it is not a square.<br />
10 Converse: If someone will trade desserts, <strong>the</strong>n<br />
I have cookies with my lunch.<br />
Inverse: If I do not have cookies with my<br />
lunch, <strong>the</strong>n someone will not trade desserts.<br />
Contrapositive: If someone will not trade<br />
desserts, <strong>the</strong>n I do not have cookies with<br />
my lunch.<br />
11 Converse: If my teacher will be able to read<br />
my paper, <strong>the</strong>n I typed it.<br />
Inverse: If I do not type my paper, <strong>the</strong>n my<br />
teacher will not be able to read it.<br />
Contrapositive: If my teacher will not be able<br />
to read my paper, <strong>the</strong>n I did not type it.<br />
12 Converse: If I am late to school, <strong>the</strong>n I didn’t<br />
set my alarm.<br />
Inverse: If I set my alarm, <strong>the</strong>n I will not be<br />
late to school.<br />
Contrapositive: If I am not late to school, <strong>the</strong>n<br />
I set my alarm.
13 Converse: If I am not in <strong>the</strong> starting lineup,<br />
<strong>the</strong>n I will not go to practice.<br />
Inverse: If I to go to practice, I will be in <strong>the</strong><br />
starting lineup.<br />
Contrapositive: If I am in <strong>the</strong> starting lineup,<br />
<strong>the</strong>n I will go to practice.<br />
14 Converse: If I am unhappy, <strong>the</strong>n I did not<br />
make honor roll.<br />
Inverse: If I make honor roll, <strong>the</strong>n I am happy.<br />
Contrapositive: If I am happy, <strong>the</strong>n I made<br />
honor roll.<br />
15 Converse: If I do not have money <strong>for</strong> <strong>the</strong><br />
movies, <strong>the</strong>n I will go shopping.<br />
Inverse: If I do not go shopping, <strong>the</strong>n I will<br />
have money <strong>for</strong> <strong>the</strong> movies.<br />
Contrapositive: If I have money <strong>for</strong> <strong>the</strong><br />
movies, <strong>the</strong>n I will not go shopping.<br />
16 Converse: If I do not take a school bus, <strong>the</strong>n I<br />
live close to school.<br />
Inverse: If I do not live close to school, <strong>the</strong>n I<br />
take a school bus.<br />
Contrapositive: If I take a school bus, <strong>the</strong>n I do<br />
not live close to school.<br />
17 Converse: If I don’t study Latin, <strong>the</strong>n I will<br />
study French.<br />
Inverse: If I don’t study French, <strong>the</strong>n I will<br />
study Latin.<br />
Contrapositive: If I study Latin, <strong>the</strong>n I will not<br />
study French.<br />
18 a If <strong>the</strong> segments are congruent, <strong>the</strong>n <strong>the</strong>ir<br />
measures are equal. (True)<br />
b If <strong>the</strong> measures of <strong>the</strong> segments are equal,<br />
<strong>the</strong>n <strong>the</strong> segments are congruent. (True)<br />
2-5 Biconditionals<br />
(pages 31–32)<br />
1 (1) true <strong>for</strong> any value of x<br />
2 (2) false <strong>for</strong> any value of x<br />
3 (2) false<br />
4 (1) If I am sleepy, <strong>the</strong>n I did not get eight<br />
hours of sleep and if I did not get eight hours<br />
of sleep, <strong>the</strong>n I am sleepy.<br />
5 (3) Paul does not buy chicken or he does not<br />
roast it.<br />
6 (4) <strong>the</strong> conjunction of a conditional and its<br />
converse<br />
7 If I have money, <strong>the</strong>n I earned it.<br />
8 If tomorrow is Friday, <strong>the</strong>n today is Thursday.<br />
(True)<br />
9 Converse: If a number is rational, <strong>the</strong>n it is a<br />
repeating decimal.<br />
Biconditional: A number is rational if and only<br />
if it is a repeating decimal.<br />
The biconditional is false <strong>for</strong> integers and<br />
terminating decimals.<br />
10 If <strong>the</strong> sum of <strong>the</strong> measures of two angles is<br />
not 45, <strong>the</strong>n <strong>the</strong> two angles are not complementary.<br />
Any pair of complementary angles makes<br />
this statement and original false.<br />
Any pair of angles with a sum of measures<br />
not equal to 45 or 90 makes this statement<br />
and <strong>the</strong> original true.<br />
2-6 Laws of Logic<br />
(page 36)<br />
1 Margaret is a doctor. (Disjunctive Inference)<br />
2 I will have straight teeth. (Detachment)<br />
3 I did not save money. (Modus Tollens)<br />
4 I helped my friend. (Modus Tollens)<br />
5 It will not rain. (Detachment)<br />
6 Joe is not a teacher. (Disjunctive Inference)<br />
7 I will not eat dinner with Michael. (Modus<br />
Tollens)<br />
8 I will not play on Saturday. (Detachment)<br />
9 If I practice <strong>the</strong> violin, my friend will be<br />
jealous. (Chain Rule)<br />
10 If I don’t help my bro<strong>the</strong>r, he cannot play<br />
football. (Chain Rule)<br />
2-7 Proof in Logic<br />
(page 38)<br />
1 1. p ∨ q<br />
2. r → ~q<br />
3. r<br />
4. ~q Law of Detachment (2, 3)<br />
5. p Law of Disjunctive Inference (1, 4)<br />
2 1. a → b<br />
2. c ∨ a<br />
3. ~c<br />
4. a Law of Disjunctive Inference (2, 3)<br />
5. b Law of Detachment (1, 4)<br />
2-7 Proof in Logic 5
3 1. e ∨ ~f<br />
2. ~f → g<br />
3. ~e<br />
4. ~f Law of Disjunctive Inference (1, 3)<br />
5. g Law of Detachment (2, 4)<br />
4 1. a ∨ b<br />
2. b → c<br />
3. c → d<br />
4. b → d Chain Rule (2, 3)<br />
5. ~d<br />
6. ~b Law of Modus Tollens (4, 5)<br />
7. a Law of Disjunctive Inference (1, 6)<br />
5 1. s ∨ t<br />
2. s → r<br />
3. r → q<br />
4. s → q Chain Rule (2, 3)<br />
5. ~q<br />
6. ~s Law of Modus Tollens (4, 5)<br />
7. t Law of Disjunctive Inference (1, 6)<br />
8. t → v<br />
9. v Law of Detachment (8, 7)<br />
6 1. d → e<br />
2. d ∨ f<br />
3. h → ~e<br />
4. h<br />
5. ~e Law of Detachment (3, 6)<br />
6. ~d Law of Modus Tollens (1, 5)<br />
7. f Law of Disjunctive Inference (2, 6)<br />
7 1. ~f → g<br />
2. ~f ∨ j<br />
3. g → ~h<br />
4. j → k<br />
5. h<br />
6. ~g Law of Modus Tollens (3, 5)<br />
7. f Law of Modus Tollens (1, 6)<br />
8. j Law of Disjunctive Inference (2, 7)<br />
9. k Law of Detachment (4, 8)<br />
8 1. x → z<br />
2. x → y<br />
3. ~z<br />
4. ~y Law of Modus Tollens (1, 3)<br />
5. ~x Law of Modus Tollens (2, 4)<br />
6. x ∨ t<br />
7. t Law of Disjunctive Inference (5, 6)<br />
Chapter Review (pages 38–40)<br />
1 (1) true<br />
2 (1) true<br />
3 (2) false<br />
6 Chapter 2: Logic<br />
4 (2) false<br />
5 (1) If I am late <strong>for</strong> school, I did not set my<br />
alarm.<br />
6 (2) If Marie is not bowling today, <strong>the</strong>n today<br />
is not Monday.<br />
7 (1) The statement is always true but its converse<br />
cannot be determined.<br />
8 Hypo<strong>the</strong>sis: The triangle has a 90 angle.<br />
Conclusion: The square of <strong>the</strong> longest side of<br />
a triangle is equal to <strong>the</strong> sum of <strong>the</strong> squares<br />
of <strong>the</strong> o<strong>the</strong>r sides.<br />
9 Hypo<strong>the</strong>sis: A and B are alternate interior<br />
angles.<br />
Conclusion: They are congruent.<br />
10 False<br />
11 True<br />
12 True<br />
13 False<br />
14 False<br />
15 Inverse: If <strong>the</strong> altitude does not bisect <strong>the</strong><br />
base, <strong>the</strong> triangle is not isosceles.<br />
Converse: If <strong>the</strong> triangle is isosceles, <strong>the</strong> altitude<br />
bisects <strong>the</strong> base.<br />
Contrapositive: If <strong>the</strong> triangle is not isosceles,<br />
<strong>the</strong> altitude does not bisect <strong>the</strong> base.<br />
Biconditional: The altitude bisects <strong>the</strong> base if<br />
and only if it is isosceles.<br />
16 True<br />
17 False<br />
18 p is true; q is false.<br />
19 True<br />
20 Converse: If a number is rational, <strong>the</strong>n it is an<br />
integer. (True)<br />
Conditional statement is true.<br />
Biconditional: A number is an integer if and<br />
only if it is rational.<br />
21 Converse: If a number is not prime, <strong>the</strong>n a<br />
number is a perfect square. (True)<br />
Conditional statement is true.<br />
Biconditional: A number is a perfect square if<br />
and only if it is not prime.<br />
22 Converse: If a parabola is tangent to <strong>the</strong><br />
x-axis, <strong>the</strong>n its roots are equal. (True)<br />
Conditional statement is true.<br />
Biconditional: The roots of a quadratic equation<br />
are equal if and only if <strong>the</strong> parabola is<br />
tangent to <strong>the</strong> x-axis.
23 Converse: If a number is real, <strong>the</strong>n it is<br />
rational. (False)<br />
Conditional statement is true.<br />
24 Converse: If x 4 √ 2 , <strong>the</strong>n x √ 32 . (True)<br />
Conditional statement is true.<br />
Biconditional: x √ 32 if and only if<br />
x 4 √ 2 .<br />
25 Converse: If a relation is a function, <strong>the</strong>n it<br />
passes <strong>the</strong> vertical line test. (True)<br />
Conditional statement is true.<br />
Biconditional: A relation passes <strong>the</strong> vertical<br />
line test if and only if it is a function.<br />
26 If I study, <strong>the</strong>n I will not get poor grades.<br />
27 If I get poor grades, <strong>the</strong>n I did not study.<br />
28 If I do not get poor grades, <strong>the</strong>n I study.<br />
29 If my baby cousin cries, <strong>the</strong>n she gets a<br />
bottle.<br />
30 If <strong>the</strong> figure is a rectangle, <strong>the</strong>n two adjacent<br />
sides <strong>for</strong>m a right angle.<br />
31 If a triangle is equilateral, <strong>the</strong>n it is equiangular.<br />
32 I like math.<br />
33 The class president is a girl. (Law of Disjunctive<br />
Inference)<br />
34 My teacher will be happy. (Law of Detachment)<br />
35 I do not go to <strong>the</strong> park. (Law of Modus<br />
Tollens)<br />
36 If I learn to knit, I will save money. (Chain<br />
Rule)<br />
37 I eat lunch. (Law of Modus Tollens)<br />
38 I feel good. (Law of Detachment)<br />
39 Rich likes lacrosse. (Law of Disjunctive<br />
Inference)<br />
40 Marlene does not babysit. (Law of Modus<br />
Tollens)<br />
41 Jack is using his computer. (Law of Modus<br />
Tollens)<br />
42 If I read novels when my teacher assigns<br />
<strong>the</strong>m, I watch less television. (Chain Rule)<br />
43 If I like school, I won’t get a job after school.<br />
(Chain Rule)<br />
44 1. a<br />
2. p ∨ t<br />
3. a → ~t<br />
4. ~t Law of Detachment (3, 1)<br />
5. p Law of Disjunctive Inference<br />
(2, 4)<br />
45 1. x → y<br />
2. ~z ∨ x<br />
3. z<br />
4. x Law of Disjunctive Inference<br />
(2, 3)<br />
5. y Law of Detachment (1, 4)<br />
46 1. c ∨ a<br />
2. ~c<br />
3. a Law of Disjunctive Inference<br />
(1, 2)<br />
4. a → b<br />
5. b Law of Detachment (4, 3)<br />
47 1. c → ~d<br />
2. d<br />
3. ~c Law of Modus Tollens (1, 2)<br />
4. ~b → c<br />
5. b Law of Modus Tollens (4, 3)<br />
6. a ∨ ~b<br />
7. a Law of Disjunctive Inference<br />
(6, 5)<br />
48 1. ~m ∨ n<br />
2. ~m → r<br />
3. r → ~q<br />
4. ~m → ~q Chain Rule (2, 3)<br />
5. q<br />
6. m Law of Modus Tollens (4, 5)<br />
7. n Law of Disjunctive Inference<br />
(1, 6)<br />
8. n → z<br />
9. z Law of Detachment (8, 7)<br />
Chapter Review 7
CHAPTER<br />
3<br />
3-1 Inductive Reasoning<br />
(page 42)<br />
1 For a polygon with n sides and vertices, from<br />
each vertex, diagonals can be drawn to n 3<br />
o<strong>the</strong>r vertices. This can happen n times. Each<br />
diagonal is counted twice from a to b and<br />
n(n 3)<br />
from b to a. There are _<br />
2 diagonals.<br />
Test: Triangle 0<br />
Quadrilateral<br />
4(4 3)<br />
_<br />
2<br />
2<br />
5(5 3)<br />
Pentagon _ 5<br />
2<br />
2 n 5<br />
1 2 3 4 5<br />
5 4 3 2 1<br />
−−−−−−−−−−−<br />
6 6 6 6 6 5(6) n(n 1) twice<br />
<strong>the</strong> sum<br />
The sum of <strong>the</strong> integers from 1 through<br />
n(n 1)<br />
n _<br />
2 .<br />
3 Answers will vary. 9, 15, 21 are some possible<br />
answers.<br />
4 Answers will vary. 3, 9, 15, 21 are some possible<br />
answers.<br />
3-2 Definitions and Logic<br />
(page 43)<br />
1 Line segments are congruent if and only if<br />
<strong>the</strong>y have <strong>the</strong> same measure.<br />
2 A point of a line segment is <strong>the</strong> midpoint if<br />
and only if it divides <strong>the</strong> line segment into<br />
two congruent segments.<br />
8 Chapter 3: Introduction to Geometric Proof<br />
Introduction to<br />
Geometric Proof<br />
3 A line is an angle bisector if and only if<br />
it divides an angle into two congruent<br />
parts.<br />
4 Lines are perpendicular if and only if <strong>the</strong><br />
lines <strong>for</strong>m right angles.<br />
5 A figure is a polygon if and only if it is a<br />
closed figure in <strong>the</strong> plane <strong>for</strong>med by three<br />
or more segments joined at <strong>the</strong>ir endpoints.<br />
3-3 Deductive Reasoning<br />
(page 44)<br />
1 1. Given<br />
2. Perpendicular lines <strong>for</strong>m right angles.<br />
3. Right triangles have one right angle.<br />
2 EG divides FEH into two adjacent angles<br />
with <strong>the</strong> same measure. FEG and HEG<br />
are congruent because <strong>the</strong>y have <strong>the</strong> same<br />
measure. An angle bisector is <strong>the</strong> ray that<br />
divides an angle into two adjacent congruent<br />
angles.<br />
3 1. Given<br />
2. Line segments equal in measure are<br />
congruent.<br />
3. A midpoint divides a line segment into<br />
two congruent segments.<br />
4 Since −−<br />
PR bisects SPQ, it divides <strong>the</strong> angle<br />
into two congruent angles. SPR QPR.<br />
If two angles are congruent, <strong>the</strong>y have <strong>the</strong><br />
same measure.
3-4 Indirect Proof<br />
(page 45)<br />
1 Statements<br />
1.<br />
Reasons<br />
−−−<br />
CD and −−−<br />
HK are<br />
not congruent.<br />
1. Assumption.<br />
2. CD HK<br />
3.<br />
2. Given.<br />
−−−<br />
CD −−−<br />
HK 3. Line segments<br />
that are equal<br />
in measure are<br />
congruent.<br />
There<strong>for</strong>e, assumption is false.<br />
2 Statements Reasons<br />
1. ABC is not a 1. Assumption.<br />
right angle.<br />
2. −−<br />
AB −−−<br />
CD 2. Given.<br />
3. ABC is a right 3. Perpendicular<br />
angle.<br />
lines <strong>for</strong>m right<br />
angles.<br />
There<strong>for</strong>e, <strong>the</strong> assumption is false.<br />
3 Statements Reasons<br />
1. ABC is not an<br />
isosceles triangle.<br />
1. Assumption.<br />
2. A B 2. Given.<br />
3. ABC is an isos- 3. An isosceles trianceles<br />
triangle. gle contains two<br />
congruent angles.<br />
There<strong>for</strong>e, <strong>the</strong> assumption is false.<br />
4 Statements<br />
1. DB is <strong>the</strong> angle bisector<br />
of ADC.<br />
Reasons<br />
1. Assumption.<br />
2. mADB <br />
mBDC<br />
2. Given.<br />
3. DB is not <strong>the</strong> bi- 3. An angle bisector<br />
sector of ADC. divides an angle<br />
into two congruent<br />
parts.<br />
There<strong>for</strong>e, <strong>the</strong> assumption is false.<br />
3-5 Postulates, Theorems,<br />
and Proof<br />
(pages 47–49)<br />
1 Yes<br />
2 No<br />
3 No<br />
4 The symmetric property of equality<br />
5 The reflexive property of congruence<br />
6 The symmetric property and transitive<br />
property of congruence<br />
7 Statements Reasons<br />
1. −−<br />
PQ −−−<br />
QR 1. Given.<br />
2. PQR is a right<br />
angle.<br />
2. Perpendicular<br />
lines are two lines<br />
that intersect to<br />
<strong>for</strong>m right angles.<br />
3. mPQR 90 3. A right angle is<br />
an angle whose<br />
degree measure<br />
is 90.<br />
4. −−<br />
XY −−<br />
YZ 4. Given.<br />
5. XYZ is a right<br />
angle.<br />
5. Perpendicular<br />
lines are two<br />
lines that intersect<br />
to <strong>for</strong>m right<br />
angles.<br />
6. mXYZ 90 6. A right angle is<br />
an angle whose<br />
degree measure<br />
is 90.<br />
7. 90 mXYZ 7. Symmetric property<br />
of equality.<br />
8. mPQR <br />
mXYZ<br />
8. Transitive property<br />
of equality.<br />
8 Statements Reasons<br />
1. AC is <strong>the</strong> angle 1. Given.<br />
bisector of BAD.<br />
2. BAC CAD 2. An angle bisector<br />
divides an angle<br />
into two congruent<br />
parts.<br />
3. mBAC <br />
mCAD<br />
4. AD is <strong>the</strong> angle<br />
bisector of CAE.<br />
3. If two angles are<br />
congruent, <strong>the</strong>y<br />
have <strong>the</strong> same<br />
measure.<br />
4. Given.<br />
5. CAD DAE 5. An angle bisector<br />
divides an angle<br />
into two congruent<br />
parts.<br />
3-5 Postulates, Theorems, and Proof 9
6. BAC DAE 6. Transitive<br />
property.<br />
7. mBAC <br />
mDAE<br />
7. If two angles are<br />
congruent, <strong>the</strong>ir<br />
measures are<br />
equal.<br />
9 Statements Reasons<br />
1. 8 x y 1. Given.<br />
2. y 3 2. Given.<br />
3. 8 x 3 3. Substitution<br />
property.<br />
4. x 3 8 4. Symmetric<br />
property.<br />
10 Statements Reasons<br />
1. M is <strong>the</strong> midpoint<br />
of −−<br />
AB .<br />
1. Given.<br />
2. −−−<br />
AM −−−<br />
MB 2. A midpoint<br />
divides a line<br />
segment into two<br />
congruent line<br />
segments.<br />
3. −−−<br />
MB −−<br />
BC 3. Given.<br />
4. −−−<br />
AM −−<br />
BC 4. Transitive<br />
property.<br />
3-6 Remaining Postulates<br />
of Equality<br />
(pages 51–52)<br />
1 Partition postulate of equality<br />
2 Division postulate of equality<br />
3 Addition postulate<br />
4 Subtraction postulate<br />
5 Division postulate of equality<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
10 Chapter 3: Introduction to Geometric Proof<br />
6 1. m1 m2<br />
2. m3 m4<br />
3. m1 m3 m2 m4<br />
4. mDAB m1 m3<br />
mBCD m2 m4<br />
5. mDAB mBCD (Substitution<br />
7 1.<br />
postulate)<br />
−−<br />
AB −−<br />
CB<br />
2. −−−<br />
AD −−<br />
CE<br />
3. AB CB<br />
4. AD CE<br />
5. AB AD CB CE<br />
6. DB EB<br />
7. −−<br />
DB −−<br />
EB<br />
8 1. AB AC<br />
2. AD <br />
(Line segments<br />
that are equal in<br />
measure are congruent.)<br />
1 _ AC<br />
3<br />
3. AE 1 _ AB<br />
3<br />
4. AD AE (Division<br />
postulate)<br />
9 1. mEAB mFBC<br />
2. AG is <strong>the</strong> angle bisector of EAB.<br />
3. BH is <strong>the</strong> angle bisector of FBC.<br />
4. m1 1 _ mEAB<br />
2<br />
5. m2 1 _ mFBC<br />
2<br />
6. m1 m2<br />
10 1. AB DE<br />
2. AC 3AB<br />
3. DF 3DE<br />
(Division<br />
postulate)<br />
4. AC DF (Multiplication<br />
postulate)<br />
Chapter Review (page 52)<br />
1 Reflexive property of equality<br />
2 Transitive property of equality<br />
3 Symmetric property of equality<br />
4 mBAD 1 _ mBAC<br />
2
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
5 1. −−−−<br />
CMD is a line segment.<br />
2. −−−<br />
CM −−−<br />
MD<br />
3. M is <strong>the</strong> mid- (Definition of a midpoint)<br />
point of −−−<br />
CD .<br />
6 1. −−−−<br />
ABCD is a line segment.<br />
2. −−<br />
AB −−−<br />
CD<br />
3. BC BC<br />
4. AB CD<br />
5. AB BC CD BC<br />
6. AC BC<br />
7. −−<br />
AC −−<br />
BD (Line segments that are<br />
equal in measure are<br />
congruent.)<br />
7 1. −−<br />
AB −−−<br />
CD<br />
2. AB CD<br />
3. −−<br />
AB and −−−<br />
CD bisect each o<strong>the</strong>r at E.<br />
4. AE 1 _ AB<br />
2<br />
5. CE 1 _ CD<br />
2<br />
6. AE CE<br />
7. −−<br />
AE −−<br />
CE (Line segments that are<br />
equal in measure are<br />
congruent.)<br />
8 1. −−<br />
PQ −−−<br />
QR<br />
2. mPQR 90<br />
3. mPQR mABC<br />
4. mABC 90<br />
5. −−<br />
AB −−<br />
BC (Perpendicular lines <strong>for</strong>m<br />
right angles.)<br />
9 1. PQ XY<br />
2. QR YZ<br />
3. PQ QR XY YZ<br />
4. PQ QR PR<br />
5. XY YZ XZ<br />
6. PR XZ<br />
7. −−<br />
PR −−<br />
XZ (Line segments that are<br />
equal in measure are<br />
congruent.)<br />
10 1. DAB and EBA are straight angles.<br />
2. mDAB 180<br />
3. mEBA 180<br />
4. m1 m2<br />
5. mDAB m1 mEBA m2<br />
6. m3 m4 (Substitution postulate)<br />
11 1. −−<br />
AC −−<br />
BC<br />
2. AC BC<br />
4. −−−<br />
DC −−<br />
EC<br />
5. DC EC<br />
6. AD DC BC EC<br />
7. AD BE<br />
8. −−−<br />
AD −−<br />
BE (Line segments that are<br />
equal in measure are<br />
congruent.)<br />
12 1. −−−<br />
AD −−<br />
BC<br />
2. AD BC<br />
3. E is <strong>the</strong> midpoint of −−−<br />
AD .<br />
4. DE 1 _ AD<br />
2<br />
5. F is <strong>the</strong> midpoint of −−<br />
BC .<br />
6. BF 1 _ BC<br />
2<br />
7. BF DE<br />
8. −−<br />
BF −−<br />
DE (Line segments that are<br />
equal in measure are<br />
congruent.)<br />
Chapter Review 11
CHAPTER<br />
4-1 Setting Up a Valid<br />
Proof<br />
(pages 56–57)<br />
1 Given: ABC , BD AC <br />
D<br />
4<br />
A C<br />
B<br />
Prove: ABD CBD<br />
Statements Reasons<br />
1. ABC <br />
2. BD AC <br />
1. Given.<br />
2. Given.<br />
3. ABD is a right<br />
angle.<br />
4. CBD is a right<br />
angle.<br />
3. Definition of perpendicular<br />
lines.<br />
4. Definition of perpendicular<br />
lines.<br />
5. ABD CBD 5. Congruent angles<br />
are angles that<br />
have <strong>the</strong> same<br />
measure.<br />
2 Given: PQRS , −−<br />
PR −−<br />
QS<br />
P Q R S<br />
Prove: −−<br />
PQ −−<br />
RS<br />
12 Chapter 4: Congruence of Lines, Angles, and Triangles<br />
Congruence of Lines,<br />
Angles, and Triangles<br />
Statements Reasons<br />
1. PQRS <br />
1. Given.<br />
2. −−<br />
PR −−<br />
QS 2. Given.<br />
3. −−−<br />
QR −−−<br />
QR 3. Reflexive<br />
property of<br />
congruence.<br />
4. PR QR <br />
QS QR<br />
4. Subtraction<br />
postulate<br />
5. −−<br />
PQ −−<br />
RS 5. Substitution<br />
postulate<br />
3 Given: −−−<br />
AD −−<br />
BC<br />
−−<br />
EF bisects −−−<br />
AD<br />
−−<br />
GF bisects −−<br />
BC<br />
Prove: −−<br />
AE −−<br />
BG<br />
Statements Reasons<br />
1. −−−<br />
AD −−<br />
BC 1. Given.<br />
2. −−<br />
EF bisects −−−<br />
AD 2. Given.<br />
3. −−<br />
GF bisects −−<br />
BC 3. Given.<br />
4. −−<br />
AE −−−<br />
AD 4. Definition of a<br />
bisector.<br />
5. −−<br />
BG −−<br />
GC 5. Definition of a<br />
bisector.<br />
6. AE ED 6. Definition of<br />
congruent<br />
segments.<br />
7. BG GC 7. Definition of<br />
congruent<br />
segments.<br />
8. AE ED AD 8. Partition<br />
postulate.
9. BG GC BC 9. Partition<br />
postulate.<br />
10. AE AE AD<br />
or 2AE AD<br />
10. Substitution<br />
postulate.<br />
11. BG BG BC or 11. Substitution<br />
2BG BC<br />
postulate.<br />
12. AE 1 _ AD<br />
2<br />
12. Division<br />
postulate.<br />
13. BG 1 _ BC<br />
2<br />
13. Division<br />
postulate.<br />
14. AD BC 14. Definition of<br />
congruent<br />
segments.<br />
15. AE 1 _ BC<br />
2<br />
15. Substitution<br />
postulate.<br />
16. 1 _ BC BG<br />
2<br />
16. Symmetric<br />
property.<br />
17. AE BG 17. Transitive<br />
property.<br />
18. −−<br />
AE −−<br />
BG 18. Segments with<br />
equal measures<br />
are congruent.<br />
4 Given: URS UTS<br />
−−<br />
US bisects RUT<br />
−−<br />
US bisects RST<br />
Prove: 1 2<br />
Statements Reasons<br />
1. URS UTS 1. Given.<br />
2. −−<br />
US bisects RUT 2. Given.<br />
3. −−<br />
US bisects RST 3. Given.<br />
4. 1 TUS 4. Definition of an<br />
angle bisector.<br />
5. RSU 2 5. Definition of an<br />
angle bisector.<br />
6. m1 mTUS 6. Congruent<br />
angles are equal<br />
in measure.<br />
7. mRSU m2 7. Congruent<br />
angles are equal<br />
in measure.<br />
8. m1 mTUS 8. Partition<br />
mRUT<br />
postulate.<br />
9. mRSU m2 9. Partition<br />
mRST<br />
postulate.<br />
10. m1 m1 <br />
mRUT or<br />
2(m1) <br />
mRUT<br />
11. m2 m2 <br />
mRST or<br />
2(m2) <br />
mRST<br />
12. m1 1 _<br />
10. Substitution<br />
postulate.<br />
11. Substitution<br />
postulate.<br />
mRUT 12. Division<br />
2<br />
postulate.<br />
13. m2 1 _ mRST 13. Division<br />
2<br />
postulate.<br />
14. mRUT <br />
mRST<br />
14. Congruent angles<br />
are equal in<br />
measure.<br />
15. m1 1<br />
_<br />
2 mRST 15. Substitution<br />
postulate.<br />
16. 1 _ mRST 2 16. Symmetric<br />
2<br />
property.<br />
17. m1 m2 17. Transitive<br />
property.<br />
18. 1 2 18. Congruent angles<br />
are equal in<br />
5 Given:<br />
measure.<br />
−−<br />
HJ −−<br />
FK<br />
Prove: −−−<br />
HK −−<br />
FJ<br />
Statements Reasons<br />
1. −−<br />
HJ −−<br />
FK 1. Given.<br />
2. HJ FK 2. Definition of congruent<br />
segments.<br />
3. JK JK 3. Reflexive<br />
property.<br />
4. HJ JK HK 4. Partition<br />
postulate.<br />
5. FK JK FJ 5. Partition<br />
postulate.<br />
6. HJ JK <br />
FK JK<br />
6. Addition<br />
postulate.<br />
7. HK FJ 7. Substitution<br />
postulate.<br />
8. −−−<br />
HK −−<br />
FJ 8. Segments with<br />
equal measures<br />
are congruent.<br />
4-1 Setting Up a Valid Proof 13
6 Given: BA bisects CBE<br />
1 2<br />
2 4<br />
Prove: 3 4<br />
Narrative Proof: A bisector divides an angle<br />
into two congruent angles, so 1 2.<br />
Using <strong>the</strong> transitive property of congruence,<br />
1 4. By substitution, 3 4.<br />
7 4x 3 2x 21<br />
x 12<br />
mBCA 4x 3<br />
4(12) 3<br />
45<br />
mBCE 2(45) 90<br />
8 4x 4 2(3x 21)<br />
x 23<br />
AB 4x 4<br />
4(23) 4<br />
96<br />
9 Since DE CE, <strong>the</strong>n 2x y 5y, and<br />
x 2y.<br />
AC BD<br />
6 5y (2x y) (x y)<br />
6 5y 3x 2y<br />
6 5y 3(2y) 2y<br />
6 5 y 6y 2y<br />
6 3y<br />
2 y<br />
x 2(2) 4<br />
BD 2(4) 2 4 2 16<br />
10 AB DE 8, so BD 4 and CD 2. There<strong>for</strong>e,<br />
CE 8 2 10.<br />
4-2 Proving Theorems<br />
About Angles<br />
(pages 59–61)<br />
1 If two angles are right angles, <strong>the</strong>y both have<br />
a measurement of 90. Angles with <strong>the</strong> same<br />
measure are congruent.<br />
2 If two angles are straight angles, <strong>the</strong>y both<br />
have a measurement of 180. Angles with <strong>the</strong><br />
same measure are congruent.<br />
3 If 1 is a complement of A, <strong>the</strong>n m1 <br />
90 mA. If 2 is a complement of A,<br />
<strong>the</strong>n m2 90 mA. Then m1 m2<br />
by <strong>the</strong> symmetric property and transition,<br />
and 1 2 by definition of congruence.<br />
14 Chapter 4: Congruence of Lines, Angles, and Triangles<br />
4 If 1 2, 3 is <strong>the</strong> complement of<br />
1, and 4 is <strong>the</strong> complement of 2, <strong>the</strong>n<br />
m3 90 m1 and m4 90 m2.<br />
Since m1 m2 by congruence, m3 <br />
m4 by symmetric property and transition,<br />
and 3 4 by congruence.<br />
5 If 1 is a supplement of A, <strong>the</strong>n m1 <br />
180 mA. If 2 is a supplement of A,<br />
<strong>the</strong>n m2 180 mA. Then m1 m2<br />
by <strong>the</strong> symmetric property and transition,<br />
and 1 2 by definition of congruence.<br />
6 If 1 2 and 3 is <strong>the</strong> supplement of<br />
1 and 4 is <strong>the</strong> supplement of 2, <strong>the</strong>n<br />
m3 90 m1 and m4 90 m2.<br />
Since m1 m2 by congruence, m3 <br />
m4 by symmetric property and transition,<br />
and 3 4 by congruence.<br />
7 By definition <strong>the</strong> sum of <strong>the</strong>ir measures is a<br />
straight angle or 180.<br />
8 The two angles are a linear pair so <strong>the</strong> sum<br />
of <strong>the</strong>ir measures is 180. If <strong>the</strong>y are congruent,<br />
each measure is 90; <strong>the</strong>y <strong>for</strong>m right<br />
angles and are <strong>the</strong>re<strong>for</strong>e perpendicular.<br />
9 Each angle <strong>for</strong>ms a linear pair with <strong>the</strong> same<br />
angle. They are supplementary to <strong>the</strong> same<br />
angle and are <strong>the</strong>re<strong>for</strong>e congruent.<br />
10 m2 m3 because <strong>the</strong>y are vertical angles.<br />
By substitution, 1 is complementary<br />
to 3. 1 4 because complements of <strong>the</strong><br />
same angle are congruent.<br />
11 3 <strong>for</strong>ms a linear pair with 1, 3 is supplementary<br />
to 1. Since 1 2, 3 is<br />
supplementary to 2. 4 <strong>for</strong>ms a linear pair<br />
with 2, 4 is supplementary to 2.<br />
3 4 because supplements of <strong>the</strong> same<br />
angle are congruent.<br />
12 Since <strong>the</strong>y <strong>for</strong>m a linear pair, 2 is supplementary<br />
to EDG. Since 1 and 2 have<br />
<strong>the</strong> same measure (<strong>the</strong>y are congruent), 1 is<br />
supplementary to EGD.<br />
13 Since 1 is complementary to 3,<br />
m1 m3 90. Since ACB is a right<br />
angle, m1 m2 90. By subtraction and<br />
substitution, m2 m3, so <strong>the</strong>y are congruent<br />
angles.<br />
14 By <strong>the</strong> transitive postulate, 1 3.<br />
Because <strong>the</strong>y are vertical angle pairs,<br />
1 4, and 3 6. By substitution<br />
and transition, 4 6.
15 x 2 5 180<br />
x 155<br />
16 2x x 180<br />
x 60<br />
17 x 85<br />
18 (x 10) x 90<br />
x 40<br />
19 60 x x x 180<br />
x 40<br />
20 6x 2 0 4x 10<br />
x 15<br />
21 40 mz 180<br />
mz 140<br />
22 a mDOE 75<br />
b mAOB 15<br />
c mDOC 105<br />
23 2x 1 5 4x 1<br />
x 7<br />
mPRT 2(7) 15 29<br />
mRTQ 180 mPRT<br />
180 29<br />
151<br />
24 3x 2 5 10x 4<br />
x 3<br />
mWVY 3(3) 25 34<br />
mWVZ 180 mWVY<br />
180 34<br />
146<br />
25 (5x 2y) (5x 2 y) 180<br />
x 18<br />
m AEC mBED<br />
11y 5x 2y<br />
1 1y 5(18) 2y<br />
9 y 90<br />
y 10<br />
mAED mCEB 5x 2y<br />
5(18) 2(10) 70<br />
4-3 Congruent Polygons<br />
(pages 64–66)<br />
1 (3) IHG MNL<br />
2 (4) a side of one is congruent to a side of <strong>the</strong><br />
o<strong>the</strong>r<br />
3 (4) IV<br />
4 (1) I<br />
5 (2) II<br />
6 a −−<br />
AC<br />
b BAC<br />
c −−<br />
BC<br />
d CBA<br />
7 a ACD BCD<br />
b ADC CBA<br />
8 a mE 44<br />
b mB 110<br />
c mC 26<br />
9 Answers will vary. For instance, a 3-4-5<br />
triangle has <strong>the</strong> same angle measure as a<br />
6-8-10 triangle. They are not congruent.<br />
10 Answers will vary.<br />
11 Answers will vary.<br />
12 a Vertical angles are congruent; <strong>the</strong> triangles<br />
are congruent by SAS.<br />
b AC BD<br />
2 x x 5<br />
x 5<br />
AC BD 10<br />
AE BE 11<br />
CE DE 12<br />
13 The triangles are congruent by SAS.<br />
ACE BDE by corresponding parts of<br />
congruent triangles are congruent. There<strong>for</strong>e,<br />
ACE BDE. So 5x 7x 18, and x 9.<br />
mACE 45; mBDE 45; mBED 45.<br />
mDBE 180 45 45 90. There<strong>for</strong>e,<br />
DBE is a right triangle.<br />
14 x 2 4x x 14<br />
x 2 5x 14 0<br />
(x 7)(x 2) 0<br />
x 7 and x 2<br />
When x 7, AB AB 21<br />
When x 2, AB AB 12<br />
15 No, <strong>the</strong> angle must be included between <strong>the</strong><br />
pairs of sides.<br />
(page 67)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
4-3 Congruent Polygons 15
1 1. −−−−<br />
ABCD<br />
2. 1 2<br />
3. ___<br />
AF −−<br />
DE<br />
4. −−<br />
AC −−<br />
BD<br />
5. AC BD<br />
6. BC BC (Reflexive property)<br />
7. AC BC BD BC<br />
8. AB CD<br />
9. −−<br />
AB −−−<br />
CD<br />
10. ABF DCE (SAS SAS)<br />
2 1. −−−<br />
ABC , −−−<br />
DEF<br />
2. 3 4<br />
3. −−<br />
BF −−<br />
EC<br />
4. 1 2<br />
5. −−−<br />
FBC −−−<br />
CEF (If two angles are<br />
congruent, <strong>the</strong>ir<br />
supplements are<br />
congruent.)<br />
6. BCF EFL (ASA ASA)<br />
3 1. −−<br />
BE is a median to −−<br />
FD .<br />
2. −−<br />
FE −−<br />
DE<br />
3. −−<br />
BE −−<br />
BE<br />
4. −−−<br />
AD −−<br />
CF<br />
5. −−<br />
AB −−<br />
CB<br />
6. AD AB CF CB<br />
7. −−<br />
BD −−<br />
BF<br />
8. FBE DBE (SSS SSS)<br />
4 1. −−<br />
AE −−−<br />
DC<br />
2. −−<br />
DE −−<br />
DE<br />
3. AE DE DC DE<br />
4. −−−<br />
AD −−<br />
EC<br />
5. 3 4<br />
6. ADB and 3 are linear pairs.<br />
7. ADB and 3 are supplements.<br />
8. CEB and 4 are linear pairs.<br />
9. CEB and 4 are supplements.<br />
10. ADB CEB<br />
11. 1 2<br />
12. ADB CEB (ASA ASA)<br />
5 1. D is <strong>the</strong> midpoint of −−<br />
AB .<br />
2. AD DB<br />
3. −−−<br />
AD −−<br />
DB<br />
4. −−<br />
AC −−<br />
BC<br />
5. −−−<br />
DC −−−<br />
DC<br />
6. ADC BDC (SSS SSS)<br />
6 1. −−<br />
AB −−<br />
BC<br />
2. −−<br />
EF −−<br />
AB<br />
3. −−<br />
EF −−<br />
BC<br />
16 Chapter 4: Congruence of Lines, Angles, and Triangles<br />
4. −−<br />
BE bisects −−<br />
CF at D.<br />
5. −−−<br />
CD −−<br />
DF<br />
6. 1 2<br />
7. BCD EFD (SAS SAS)<br />
7 1. −−<br />
AC −−<br />
BD<br />
2. −−<br />
BC −−<br />
BC<br />
3. AC BC BD BC<br />
4. −−<br />
AB −−−<br />
DC<br />
5. 3 4<br />
6. 3 and FBA are linear pairs.<br />
7. 3 and FBA are supplements.<br />
8. 4 and ECD are linear pairs.<br />
9. 4 and ECD are supplements.<br />
10. FBA ECD<br />
11. 1 2<br />
12. EDC FAB (ASA ASA)<br />
8 1. EG is <strong>the</strong> perpendicular bisector of −−<br />
AB .<br />
2. −−<br />
AF −−<br />
BF<br />
3. EFA is a right angle.<br />
4. EFB is a right angle.<br />
5. EFA EFB<br />
6. mEFA m1 mEFB m1<br />
7. 1 2<br />
8. mEFA m1 mEFB m2<br />
9. DFA CFB<br />
10. −−<br />
DF −−<br />
CF<br />
11. ADF BCF (SAS SAS)<br />
Chapter Review (pages 68–69)<br />
1 Given: ABC and DBE are vertical angles.<br />
PQR ABC<br />
Prove: PQR DBE<br />
2 Given: AB and CD intersect at E.<br />
AEC FGH<br />
Prove: CEB is supplementary to FGH.<br />
3 Given: AEB and CED are perpendicular lines.<br />
Prove: AEC AED<br />
4 Given: AEB and CED are perpendicular lines.<br />
F is not on CD .<br />
Prove: FE is not perpendicular to AB .<br />
5 Since E is <strong>the</strong> midpoint of −−<br />
AB , CD is a bisector<br />
of −−<br />
AB . Since AEC BEC and <strong>the</strong>y are<br />
a linear pair, <strong>the</strong> measure of each must be<br />
180 _ 90.<br />
2
6 1 3 because <strong>the</strong>y are vertical angles.<br />
4 2 because <strong>the</strong>y are vertical angles. By<br />
<strong>the</strong> substitution postulate, 2 3. By <strong>the</strong><br />
transitive postulate, 4 3, or 3 4<br />
by <strong>the</strong> symmetric property.<br />
7 Since −−<br />
PQ bisects ABC, <strong>the</strong>n 2 CBQ.<br />
Since CBQ and 1 are vertical pairs,<br />
CBQ 1. By <strong>the</strong> transitive postulate,<br />
2 1, or 1 2 by <strong>the</strong> symmetric<br />
property.<br />
8 Since CDE is a right triangle, <strong>the</strong> two angles<br />
that are not <strong>the</strong> right angle are complementary<br />
because <strong>the</strong>re sum is 180 (whole<br />
triangle) 90 (right angle) 90. There<strong>for</strong>e,<br />
FED is complementary to FCD. By substitution,<br />
EDF is complementary to FCD.<br />
9 mAED mBED 180<br />
4x 20 9x 4 8 180<br />
x 16<br />
mCEB mAED<br />
4x 20<br />
4(16) 20<br />
84<br />
10 mAED mCEB<br />
5 x 7x 26<br />
x 13<br />
mAED 5(13) 65<br />
mAED mDEB 180<br />
65 mDEB 180<br />
mDEB 115<br />
11 mAED mCEB<br />
1 2x 6x 10y<br />
6x 10y<br />
mAED mDEB 180<br />
12x 6x 180<br />
18 x 180<br />
x 10<br />
6x 10y<br />
6 0 10y<br />
y 6<br />
mAEC mDEB<br />
10y<br />
10(6) 60<br />
12 mCEB mDEB 180<br />
9y 12y 1 2 180<br />
y 8<br />
mAEC mDEB<br />
12(8) 12 108<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
13 1. −−<br />
AC −−<br />
DF<br />
2. A D<br />
3. C F<br />
4. ABC DEF (ASA ASA)<br />
14 1. −−<br />
HE −−<br />
FE<br />
2. −−−<br />
HG −−<br />
FE<br />
3. −−<br />
HF −−<br />
HF<br />
4. EFH GHF (SSS SSS)<br />
15 1. A, B, C, D lie on circle O.<br />
2. −−−−<br />
AOC ; −−−−<br />
BOD<br />
3. −−−<br />
AO , −−<br />
BO , −−−<br />
CO , −−−<br />
DO are radii.<br />
4. −−−<br />
AD −−−<br />
CO<br />
5. −−<br />
BO −−−<br />
DO<br />
6. AOB COD (If two lines<br />
intersect, <strong>the</strong><br />
vertical angles are<br />
congruent.)<br />
7. ABO CDO (SAS SAS)<br />
16 1. −−<br />
QT −−<br />
RT<br />
2. QT RT<br />
3. −−<br />
TV −−<br />
TU<br />
4. TV TU<br />
5. QT TV RT TV<br />
6. QT TV RT TU<br />
7. −−−<br />
QTV ; −−−<br />
RTU<br />
8. QU RU<br />
9. −−−<br />
QV −−−<br />
RU<br />
10. −−<br />
PU −−<br />
VS<br />
11. −−<br />
PU −−−<br />
UV −−<br />
VS −−−<br />
UV<br />
12. −−<br />
PV −−<br />
SU<br />
13. −−<br />
PQ −−<br />
RS<br />
14. PQU SRU (SSS SSS)<br />
17 1. C is <strong>the</strong> midpoint of −−<br />
AE .<br />
2. −−<br />
AC −−<br />
CE<br />
3. 1 2<br />
4. 3 BCD<br />
5. 4 BCD<br />
6. 3 4<br />
7. ABC EDC (ASA ASA)<br />
Chapter Review 17
18 1. I is <strong>the</strong> midpoint of −−<br />
EG .<br />
2. −−<br />
EI −−<br />
IG<br />
3. EI IG<br />
4. −−<br />
EG −−<br />
EI −−<br />
IG<br />
5. EG EI IG<br />
6. EG EI EI or EG 2EI<br />
7. EH 2EI<br />
8. EG EH<br />
9. −−<br />
EG −−<br />
EH<br />
10. FGE IEH<br />
11. FEG HIE<br />
12. FGE IEH (ASA ASA)<br />
5-1 Line Segments<br />
Associated With Triangles<br />
(page 71)<br />
1 One line<br />
2 C<br />
Median<br />
CHAPTER<br />
5<br />
A E F D B<br />
Angle Bisector<br />
3 −−<br />
AE and −−<br />
BE<br />
4 −−<br />
AB and −−−<br />
CD<br />
5 ACF and BCF<br />
6 ADC and BDC<br />
Altitude<br />
18 Chapter 5: Congruence Based on Triangles<br />
19 1. AEB, CED are right angles.<br />
2. AEB CED<br />
3. A C<br />
4. −−<br />
AE −−<br />
CE<br />
5. ABC CDE (ASA ASA)<br />
20 1. −−−<br />
DC −−−<br />
AD<br />
2. −−<br />
AB −−−<br />
AD<br />
3. −−−<br />
DC −−<br />
AB<br />
4. −−<br />
DF −−<br />
BE<br />
5. −−<br />
EF −−<br />
EF<br />
6. DF EF BE EF<br />
7. −−<br />
DE −−<br />
BF<br />
8. 1 2<br />
9. ABF DCE (SAS SAS)<br />
Congruence Based on<br />
Triangles<br />
5-2 Using Congruent<br />
Triangles to Prove Line<br />
Segments Congruent and<br />
Angles Congruent<br />
(pages 72–73)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)
1 1. −−<br />
AB −−−<br />
CD<br />
2. −−<br />
BC −−−<br />
DA<br />
3. −−<br />
AC −−<br />
AC<br />
4. ABC CDA (SSS SSS)<br />
5. BAC DCA (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
2 1. −−<br />
BA −−<br />
BC<br />
2. −−−<br />
DA −−−<br />
DC<br />
3. −−<br />
DB −−<br />
DB<br />
4. ABD CBD (SSS SSS)<br />
5. ABC CBD (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
3 1. 1 3<br />
2. 2 4<br />
3. −−<br />
AC −−<br />
AC<br />
4. DAC BCA (ASA ASA)<br />
5. −−−<br />
AD −−<br />
CB (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
4 1. −−−<br />
CD is <strong>the</strong> median drawn from C.<br />
2. −−−<br />
AD −−<br />
DB<br />
3. −−−<br />
CD −−<br />
AB<br />
4. ADC is a right angle.<br />
5. BCD is a right angle.<br />
6. ADC BDC<br />
7. −−−<br />
CD −−−<br />
CD<br />
8. ADC BDC (SAS SAS)<br />
9. −−<br />
CA −−<br />
CB<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
5 RS RS<br />
4x 1 3x 3<br />
x 4<br />
RT RT x 6 4 6 10<br />
6 AD CB<br />
2x 5 3x 7<br />
x 12<br />
AB x 10 12 10 22<br />
CD AB 22<br />
AD 2x 5 2(12) 5 29<br />
CB 3x 7 3(12) 7 29<br />
7 The triangle is isosceles and RS RT.<br />
6x 4 3x 11<br />
x 5<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
8 1. 1 3<br />
2. 2 4<br />
3. −−<br />
AB −−−<br />
CD<br />
4. ABE CDE (ASA ASA)<br />
5. −−<br />
AE −−<br />
CE<br />
6. −−<br />
BD bisects −−<br />
AC .<br />
7. −−<br />
BE −−<br />
DE<br />
8. −−<br />
AC bisects −−<br />
BD . (Definition of a<br />
bisector)<br />
9 1. −−−<br />
DA bisects BDF.<br />
2. FDA BDA<br />
3. 1 2<br />
4. m1 mFDA m2 mFDA<br />
5. m1 mFDA m2 mBDA<br />
6. EDA CDA<br />
7. −−−<br />
CD −−<br />
DE<br />
8. −−−<br />
AD −−−<br />
AD<br />
9. EDA CDA (SAS SAS)<br />
10. −−<br />
AE −−<br />
AC (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
10 1. −−<br />
BA is a median of CBF.<br />
2. −−<br />
CA −−<br />
FA<br />
3. −−−<br />
CD −−<br />
FE<br />
4. −−<br />
FC −−−<br />
CD<br />
5. −−<br />
FC −−<br />
EF<br />
6. DCA is a right angle.<br />
7. EFA is a right angle.<br />
8. DCA EFA<br />
9. DCA EFA (SAS SAS)<br />
10. −−−<br />
DA −−<br />
EA (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
5-2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent 19
5-3 Isosceles and<br />
Equilateral Triangles<br />
(page 74)<br />
1 Statements Reasons<br />
1. Construct a<br />
median from <strong>the</strong><br />
vertex <strong>for</strong>med<br />
by <strong>the</strong> two congruent<br />
sides of<br />
<strong>the</strong> triangle, to<br />
<strong>for</strong>m RST with<br />
median −−−<br />
RU .<br />
1. A median of a<br />
triangle is a line<br />
segment with one<br />
endpoint at any<br />
vertex of <strong>the</strong><br />
triangle, extending<br />
to <strong>the</strong><br />
midpoint of <strong>the</strong><br />
opposite side.<br />
2. −−<br />
RS −−<br />
RT 2. Definition of an<br />
isosceles triangle.<br />
3. −−<br />
SU −−<br />
TU 3. Definition of a<br />
median.<br />
4. −−−<br />
RU −−−<br />
RU 4. Reflexive<br />
property of<br />
congruence.<br />
5. RSU RTU 5. SSS SSS.<br />
6. RSU RTU 6. Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.<br />
2 Statements Reasons<br />
1. Construct a<br />
median from <strong>the</strong><br />
vertex <strong>for</strong>med<br />
by <strong>the</strong> two congruent<br />
sides of<br />
<strong>the</strong> triangle, to<br />
<strong>for</strong>m RST with<br />
median −−−<br />
RU .<br />
1. A median of a<br />
triangle is a line<br />
segment with one<br />
endpoint at any<br />
vertex of <strong>the</strong> triangle,<br />
extending<br />
to <strong>the</strong> midpoint of<br />
<strong>the</strong> opposite side.<br />
2. −−<br />
RS −−<br />
RT 2. Definition of an<br />
isosceles triangle.<br />
3. −−<br />
SU −−<br />
TU 3. Definition of a<br />
median.<br />
4. −−−<br />
RU −−−<br />
RU 4. Reflexive property<br />
of congruence.<br />
5. RSU RTU 5. SSS SSS.<br />
20 Chapter 5: Congruence Based on Triangles<br />
6. SRU TRU 6. Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.<br />
7. −−−<br />
RU bisects SRT. 7. Definition of<br />
angle bisector.<br />
3 Statements Reasons<br />
1. Construct a<br />
median from <strong>the</strong><br />
vertex <strong>for</strong>med<br />
by <strong>the</strong> two congruent<br />
sides of<br />
<strong>the</strong> triangle, to<br />
<strong>for</strong>m RST with<br />
median −−−<br />
RU .<br />
1. A median of a<br />
triangle is a line<br />
segment with one<br />
endpoint at any<br />
vertex of <strong>the</strong><br />
triangle, extending<br />
to <strong>the</strong><br />
midpoint of <strong>the</strong><br />
opposite side.<br />
2. −−<br />
RS −−<br />
RT 2. Definition of an<br />
isosceles triangle.<br />
3. −−<br />
SU −−<br />
TU 3. Definition of a<br />
median.<br />
4. −−−<br />
RU −−−<br />
RU 4. Reflexive property<br />
of congruence.<br />
5. RSU RTU 5. SSS SSS.<br />
6. SUR TUR 6. Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.<br />
7. SUR is a right<br />
angle; TUR is a<br />
right angle.<br />
7. Adjacent congruent<br />
angles are<br />
supplementary.<br />
8. −−−<br />
RU −−<br />
ST 8. Definition of perpendicular<br />
lines.<br />
4 Statements Reasons<br />
1. Construct a<br />
median from <strong>the</strong><br />
vertex <strong>for</strong>med<br />
by <strong>the</strong> two congruent<br />
sides of<br />
<strong>the</strong> triangle, to<br />
<strong>for</strong>m ABC with<br />
median −−−<br />
AX .<br />
1. A median of a<br />
triangle is a line<br />
segment with<br />
one endpoint at<br />
any vertex of<br />
<strong>the</strong> triangle,<br />
extending to <strong>the</strong><br />
midpoint of <strong>the</strong><br />
opposite side.<br />
2. −−<br />
AB −−<br />
AC 2. Definition of<br />
an equilateral<br />
triangle.
3. −−<br />
BX −−<br />
XC 3. Definition of a<br />
median.<br />
4. −−−<br />
AX −−−<br />
AX 4. Reflexive<br />
property of<br />
congruence.<br />
5. ABX ACX 5. SSS SSS.<br />
6. ABX ACX 6. Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.<br />
7. Construct a second<br />
median, −−<br />
BY ,<br />
from vertex B.<br />
7. A median of a<br />
triangle is a line<br />
segment with<br />
one endpoint<br />
at any vertex<br />
of <strong>the</strong> triangle,<br />
extending to <strong>the</strong><br />
midpoint of <strong>the</strong><br />
opposite side.<br />
8. −−<br />
BA −−<br />
BC 8. Definition of an<br />
equilateral<br />
triangle.<br />
9. −−<br />
AY −−<br />
YC 9. Definition of a<br />
median.<br />
10. −−<br />
BY −−<br />
BY 10. Reflexive<br />
property of<br />
congruence.<br />
11. BAY BCY 11. SSS SSS.<br />
12. BAY BCY 12. Corresponding<br />
angles of congruent<br />
triangles are<br />
congruent.<br />
13. ABC is equiangular.<br />
13. BCY is <strong>the</strong><br />
same as ACX,<br />
and all three<br />
angles of <strong>the</strong> triangle<br />
are equal.<br />
5 mA mC 3x 5<br />
mA mB mC 180<br />
(3x 5) (4x 10) (3x 5) 180<br />
10 x 180<br />
x 18<br />
mC 3(18) 5 59<br />
6 The measure of each angle is 60. So,<br />
3x 6 60; x 22. And 3y 6 60;<br />
y 18.<br />
7 Show that RXZ TYZ SYX by<br />
SAS SAS, and −−<br />
XY −−<br />
YZ −−<br />
ZX because<br />
corresponding parts of congruent triangles<br />
are congruent.<br />
8 Construct −−<br />
AC and −−<br />
BD , and label <strong>the</strong>ir intersection<br />
E. ABE CBE. −−<br />
AC and −−<br />
BD are<br />
perpendicular. AED and CDE are right<br />
angles; <strong>the</strong>re<strong>for</strong>e, AED CDE.<br />
−−<br />
AE −−<br />
CE . −−<br />
ED −−<br />
ED . AED CED.<br />
CAD ACD.<br />
9 mA mB mC 180<br />
(6x 12) (8x 8) (3x 6) 180<br />
17x 1 0 180<br />
x 10<br />
mA 72; mB 72; mC 36<br />
The triangle is isosceles.<br />
10 mB 180 2x<br />
5-4 Working With Two<br />
Pairs of Congruent<br />
Triangles<br />
(pages 75–76)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. ABC EFG<br />
2. −−<br />
AC −−<br />
EG<br />
3. −−<br />
BD is a median.<br />
4. D is <strong>the</strong> midpoint (Definition of<br />
of −−<br />
AC . median)<br />
5. DC 1 _ AC (Definition of mid-<br />
2<br />
point)<br />
6. −−<br />
FH is a median.<br />
7. H is <strong>the</strong> midpoint of −−<br />
EG .<br />
8. HG 1 _ EG<br />
2<br />
9. AC EG<br />
10. HG 1 _ AC<br />
2<br />
11. HG DC<br />
5-4 Working With Two Pairs of Congruent Triangles 21
12. −−−<br />
HG −−−<br />
DC<br />
13. −−<br />
FG −−<br />
BC<br />
14. BCD FGH<br />
15. BCD FGH (SAS SAS)<br />
16. −−<br />
BD −−<br />
FH<br />
2 a 1. −−<br />
PR and −−<br />
QS bisect each o<strong>the</strong>r at V.<br />
2. −−<br />
PV −−<br />
RV<br />
3. −−−<br />
QV −−<br />
SV<br />
4. PVQ RVS (Vertical angles are<br />
congruent.)<br />
5. PQV RSV (SAS SAS)<br />
b A continuation of part a.<br />
6. UPV TRV (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
7. PVU RVT (Vertical angles are<br />
congruent.)<br />
8. PUV RTV<br />
3 a 1.<br />
(ASA ASA)<br />
−−<br />
AC −−−<br />
AD<br />
2. −−<br />
BC −−<br />
BD<br />
3. −−<br />
AB −−<br />
AB<br />
4. ABC ABD (SSS SSS)<br />
5. BAC BAD (Corresponding<br />
parts of congruent<br />
triangles are<br />
6.<br />
congruent.)<br />
−−<br />
AE −−<br />
AE<br />
7. ACD ADE (SAS SAS)<br />
b A continuation of part a.<br />
8. 1 2 (Corresponding<br />
parts of congruent<br />
triangles are<br />
4 1.<br />
congruent.)<br />
−−−<br />
CD −−<br />
CB<br />
2. 1 2<br />
3. −−<br />
CF −−<br />
CF<br />
4. CFB CFD (SAS SAS)<br />
5. CBF CDF<br />
6. EDF ABF<br />
7. BFA EFD<br />
8.<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
−−<br />
DF −−<br />
BF<br />
9. BFA DFE (ASA ASA)<br />
10. 3 4 (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
22 Chapter 5: Congruence Based on Triangles<br />
5 1. −−<br />
AC −−<br />
BC<br />
2. −−<br />
AE −−<br />
BE<br />
3. −−<br />
EC −−<br />
EC<br />
4. ACE BCE (SSS SSS)<br />
5. ACE BCE<br />
6. −−−<br />
CD −−−<br />
CD<br />
7. ACD BCD (SAS SAS)<br />
8. −−−<br />
AD −−<br />
BD<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
6 a 1. −−<br />
SP −−−<br />
QR<br />
2. 1 2<br />
3. −−<br />
AB bisects −−<br />
QS at C.<br />
4. −−<br />
SC −−−<br />
QC (Definition of<br />
bisector)<br />
5. −−<br />
SQ −−<br />
SQ<br />
6. SQP QSR (SAS SAS)<br />
7. SQP QSR (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
8. ACQ BCS<br />
9. ACQ BCS (ASA ASA)<br />
b Continuation of part a.<br />
10. −−<br />
AC −−<br />
BC (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
5-5 Proving Overlapping<br />
Triangles Congruent<br />
(page 77)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. −−<br />
AE −−<br />
BE<br />
2. EAB EBA (If two sides of<br />
a triangle are congruent,<br />
<strong>the</strong> angles<br />
opposite <strong>the</strong>se sides<br />
are congruent.)
3. −−<br />
AB −−<br />
AB<br />
4. −−<br />
AC −−<br />
BD<br />
5. ABC BAD<br />
2 1. 1 3<br />
2. 2 2<br />
3. 1 3 2 3<br />
(SAS SAS)<br />
4. ADB EDC<br />
5. 4 5<br />
6.<br />
(Partition<br />
postulate)<br />
−−−<br />
AD −−<br />
ED<br />
7. ADB EDC (ASA ASA)<br />
3 1. −−<br />
AE −−<br />
BE<br />
2. D is <strong>the</strong> midpoint of −−<br />
AB .<br />
3. −−−<br />
AD −−<br />
BD<br />
4. −−<br />
ED −−<br />
ED<br />
5. ADE BDE<br />
4 1. −−<br />
TP −−<br />
TQ<br />
2. TPQ TQP<br />
3. SPQ RQP<br />
4. SPT RQT<br />
5. STP RTQ (Vertical angles are<br />
congruent.)<br />
6. STP RTQ<br />
7.<br />
(SAA SAA)<br />
−−<br />
SP −−<br />
PQ<br />
8. −−<br />
PQ −−<br />
PQ<br />
9. SPQ RQP<br />
5 1. −−<br />
EF −−<br />
GF<br />
2. −−<br />
DF −−<br />
HF<br />
3. DFH DFH<br />
4. DFG EFH (SAS SAS)<br />
5. 1 2 (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
6 1. −−−<br />
AD and −−<br />
FC bisect each o<strong>the</strong>r at G.<br />
2. −−−<br />
AG −−−<br />
DG<br />
3. −−<br />
FG −−<br />
CG<br />
4. AGC DGF (Vertical angles are<br />
congruent.)<br />
5. AGC DGF (SAS SAS)<br />
6. GDE GAC<br />
7. AGB DGE<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
8. AGB DGE (ASA ASA)<br />
5-6 Perpendicular<br />
Bisectors of a Line<br />
Segment<br />
(pages 79–80)<br />
1 Given −−<br />
AB and P and Q, such that PA PB<br />
and QA QB, let −−<br />
AB intersect −−<br />
PQ at<br />
point M. APQ BPQ. APM BPM.<br />
APM BPM. −−−<br />
AM −−−<br />
BM . There<strong>for</strong>e,<br />
AMP BMP.<br />
2 a) Given −−<br />
AB and point P such that PA PB;<br />
if two points are each equidistant from <strong>the</strong><br />
endpoints of a line segment, <strong>the</strong>n <strong>the</strong> points<br />
determine <strong>the</strong> perpendicular bisector of <strong>the</strong><br />
line segment. Select a second point that is<br />
equidistant from both A and B, <strong>for</strong> example<br />
<strong>the</strong> midpoint of −−<br />
AB , M. Then −−−<br />
PM is <strong>the</strong> perpendicular<br />
bisector of −−<br />
AB .<br />
b) Given point P on <strong>the</strong> perpendicular bisector<br />
of −−<br />
AB , let point M be <strong>the</strong> midpoint of −−<br />
AB .<br />
PMA PMB.<br />
3 The perpendicular bisector of −−<br />
PQ<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
4 1. −−<br />
JL −−<br />
KL<br />
2. −−<br />
JM −−−<br />
KM<br />
3. JL KL (Congruent segments are<br />
equal in length.)<br />
4. JM KM<br />
5. −−−<br />
LM is <strong>the</strong> perpendicular bisector of −−<br />
JK .<br />
(Two points, each equidistant from <strong>the</strong><br />
endpoints of a line segment, determine<br />
<strong>the</strong> perpendicular bisector of <strong>the</strong> line<br />
segment.)<br />
6. N is <strong>the</strong> midpoint of JK.<br />
(A point on <strong>the</strong> perpendicular bisector<br />
of a line segment is equidistant from <strong>the</strong><br />
endpoints of <strong>the</strong> line segment.)<br />
7. −−<br />
JN −−−<br />
KN (Definition of a midpoint)<br />
5-6 Perpendicular Bisectors of a Line Segment 23
5 1. −−<br />
FG is <strong>the</strong> perpendicular bisector of −−<br />
HI .<br />
2. −−<br />
JG is <strong>the</strong> perpendicular bisector of −−<br />
HI .<br />
(F, J, and G are collinear.)<br />
3. −−<br />
HJ −<br />
IJ<br />
4. −−<br />
GJ −−<br />
GJ<br />
5. HJG and IJG are right angles.<br />
(Definition of perpendicular bisector)<br />
6. HJG IJG (Right angles are<br />
congruent.)<br />
7. HGJ IGJ (SAS SAS)<br />
8. 1 2 (Corresponding parts of<br />
congruent triangles are congruent.)<br />
6 1. −−<br />
AB and −−−<br />
CD bisect each o<strong>the</strong>r.<br />
2. −−<br />
AB −−−<br />
CD<br />
3. −−<br />
AC −−−<br />
AD<br />
4. −−<br />
BC −−<br />
BD<br />
5. −−−<br />
AD −−<br />
BD<br />
6. −−<br />
AC −−<br />
BC<br />
7. ACBD is an equilateral quadrilateral.<br />
(Definition of equilateral quadrilateral)<br />
7 1. 1 2<br />
2. 3 4<br />
3. −−<br />
XZ −−<br />
XZ<br />
4. XVZ XWZ (ASA ASA)<br />
5. −−−<br />
XV −−−<br />
XW (Corresponding parts<br />
of congruent triangles<br />
are congruent.)<br />
6. XV XW<br />
7. −−−<br />
XZY ; −−−−<br />
VYW<br />
8. −−<br />
XY is <strong>the</strong> perpendicular bisector of −−−−<br />
VYW .<br />
(A point equidistant from <strong>the</strong> endpoints<br />
of a line segment is on <strong>the</strong> perpendicular<br />
bisector of <strong>the</strong> line segment.)<br />
8 BP CP<br />
4 y 4<br />
y 1<br />
AP CP<br />
x y 4<br />
x 1 4<br />
x 5<br />
9 AP BP<br />
3x y x y<br />
x y<br />
BP CP<br />
x y 4<br />
x x 4<br />
x 2<br />
y 2<br />
10 CG AG 10<br />
24 Chapter 5: Congruence Based on Triangles<br />
5-7 Constructions<br />
(page 83)<br />
1 (4) BAC and GHI<br />
2 A B C<br />
3<br />
4<br />
5<br />
S<br />
A B<br />
V<br />
E F<br />
T<br />
C D<br />
R<br />
1 2<br />
3<br />
6 If <strong>the</strong> legs of <strong>the</strong> compass <strong>for</strong>m two sides of<br />
a triangle, <strong>the</strong>n <strong>the</strong> line segment connecting<br />
<strong>the</strong> pencil to <strong>the</strong> point is <strong>the</strong> third side. If<br />
<strong>the</strong> angle between <strong>the</strong> legs does not change<br />
(and <strong>the</strong> legs remain <strong>the</strong> same length), <strong>the</strong>n<br />
any line segment connecting <strong>the</strong> legs is<br />
congruent to <strong>the</strong> original because of SAS<br />
congruence.<br />
7<br />
A<br />
G<br />
F<br />
B<br />
E<br />
H C<br />
D L<br />
The first arc drawn in <strong>the</strong> construction creates<br />
an isosceles triangle, BGH. The rest of<br />
<strong>the</strong> construction copies <strong>the</strong> sides of BGH to<br />
a new triangle. ED BH, FD GH, and<br />
FE GB, since F and G lie on <strong>the</strong> intersections.<br />
There<strong>for</strong>e, BGH EFD by SSS<br />
congruence and all corresponding angle<br />
measures, including B and E, are equal.
8 Draw a point. Draw an arc centering on <strong>the</strong><br />
point. Draw two lines from <strong>the</strong> point intersecting<br />
<strong>the</strong> arc. Connect <strong>the</strong> points of intersection.<br />
This will <strong>for</strong>m an isosceles triangle.<br />
9<br />
E<br />
10<br />
11<br />
12<br />
A P<br />
C D<br />
B<br />
A<br />
D<br />
B C<br />
A<br />
T<br />
C D<br />
A B<br />
B<br />
M<br />
C<br />
13<br />
14<br />
15<br />
16<br />
A<br />
A<br />
B<br />
A<br />
A<br />
D<br />
D<br />
C<br />
D<br />
D<br />
B<br />
C<br />
C<br />
B<br />
C<br />
B<br />
5-7 Constructions 25
Chapter Review (pages 84–85)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable statements<br />
to complete each proof. Depending on <strong>the</strong><br />
textbook used, <strong>the</strong> wording and <strong>for</strong>mat of reasons<br />
may differ, so <strong>the</strong>y have not been supplied <strong>for</strong> <strong>the</strong><br />
method of congruence applied in each problem.<br />
(These solutions are intended to be used as a<br />
guide—o<strong>the</strong>r possible solutions may vary.)<br />
1 1. −−−<br />
MN bisects PNQ.<br />
2. RND RNQ<br />
3. RNP RQN<br />
4. −−−<br />
RN −−−<br />
RN<br />
5. RPN RQN (ASA ASA)<br />
2 1. −−<br />
PQ −−<br />
PR<br />
2. −−<br />
QT is a median.<br />
3. −−<br />
PT −−<br />
RT<br />
4. −−<br />
RS is a median.<br />
5. −−<br />
PS −−<br />
QS<br />
6. PQT PRS (SSS SSS)<br />
3 1. 3 4<br />
2. −−<br />
DE −−<br />
DF<br />
3. −−−<br />
DC −−−<br />
DC<br />
4. DEC DFC (SAS SAS)<br />
5. DCE DCF (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
6. EDC FDC<br />
7. mCAD m1 mEDC mDCE<br />
180<br />
8. mCBD m2 mFDC mDCF<br />
180<br />
9. mCAD mCBD<br />
10. CAD CBD<br />
11. ABC is isosceles. (Definition of isosceles<br />
triangle)<br />
4 1. ABC is an equilateral triangle.<br />
2. ___<br />
AB ___<br />
AC (Definition of equilateral<br />
triangle)<br />
3. DCB DBC<br />
4. DB DC<br />
5. ___<br />
AD ___<br />
AD<br />
6. ABD ACD (SSS SSS)<br />
7. BAD CAD (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
8. −−−<br />
AD bisects BAC. (Definition of an<br />
angle bisector)<br />
26 Chapter 5: Congruence Based on Triangles<br />
5 1. ____<br />
TM ___<br />
TA<br />
2. MTA is isosceles. (Definition of isosceles<br />
triangle)<br />
3. ___<br />
TH bisects MTA.<br />
4. ___<br />
TH is an altitude (In an isosceles<br />
of MTA. triangle, <strong>the</strong><br />
bisector is <strong>the</strong><br />
altitude.)<br />
6 1. ___<br />
AB ___<br />
AD<br />
2. ___<br />
CB ___<br />
CD<br />
3. ___<br />
AC ___<br />
AC<br />
4. ABC ADC (SSS SSS)<br />
5. BAE DAE (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent)<br />
6. ___<br />
AE ___<br />
AE<br />
7. ABE ADE (ASA ASA)<br />
8. ___<br />
BE ___<br />
DE<br />
9. DAB is isoceles. (Definition of<br />
isosceles triangle)<br />
10. ___<br />
AE is <strong>the</strong> median (Definition of<br />
of DAB. median)<br />
11. ___<br />
AE is <strong>the</strong> altitude (In an isosceles triof<br />
DAB. angle, <strong>the</strong> median<br />
is <strong>the</strong> altitude.)<br />
7 1. ___<br />
AB ___<br />
BD<br />
2. A D<br />
3. DBA CBE<br />
4. EBD EBD<br />
5. DBA EBD CBE EBD<br />
6. EBA CBD<br />
7. ABE DBC (ASA ASA)<br />
8 1. ADE BDC<br />
2. EDB EDB<br />
3. ADE EDB BDC EDB<br />
4. ADB EDC<br />
5. DAE DEC<br />
6. ___<br />
DA ___<br />
DE<br />
7. DAB DEC (ASA ASA)<br />
9 1. 1 2<br />
2. 3 4<br />
3. ___<br />
DE ___<br />
DF<br />
4. DEG DFH (ASA ASA)<br />
5. ___<br />
GD ____<br />
HD (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
6. EDF EDF
7. 1 EDF 2 EDF<br />
8. GDF EDH<br />
9. FGD EDH (ASA ASA)<br />
10. 5 6 (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
10 1. ___<br />
AC and ___<br />
BD bisect each o<strong>the</strong>r at G.<br />
2. ___<br />
CG ____<br />
AG<br />
3. ___<br />
BG ___<br />
DG<br />
4. AGB CGD (Vertical angles are<br />
congruent.)<br />
5. AGB CGD<br />
6.<br />
(ASA ASA)<br />
___<br />
AB ___<br />
CD<br />
7. GCD GAB<br />
8. 1 2<br />
(Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
9. CED AFB<br />
10.<br />
(ASA ASA)<br />
___<br />
EC ___<br />
FA (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
11 Assume that ABC and ABC are congruent<br />
and that ___<br />
BD and _____<br />
BD are angle bisectors<br />
of B and B, respectively.<br />
ABD ABD. ___<br />
AB ____<br />
AB . A A.<br />
ABD ABC. ___<br />
BD _____<br />
BD .<br />
12 1. BIG is equilateral.<br />
2. ___<br />
IA ___<br />
BC ___<br />
GT<br />
3. IA BC GT<br />
4. __<br />
IB ___<br />
BG ___<br />
GI<br />
5. IB BG GI<br />
6. IB IA AB; BG BC CG;<br />
GI GT TI<br />
7. IA AB BC CG GT TI<br />
8. IA AB IA BC CG BC <br />
GT TI GT<br />
9. AB CG TI<br />
10. ___<br />
AB ___<br />
CG __<br />
TI<br />
11. BIG IGB GBI<br />
12. IAT BCA (ASA ASA)<br />
GTC<br />
13. ___<br />
AT ___<br />
CA ___<br />
TC (Corresponding<br />
parts of congruent<br />
triangles are<br />
congruent.)<br />
14. CAT is (Definition of an<br />
equilateral. equilateral triangle)<br />
13 1. ___<br />
RU<br />
2. ___<br />
RT ___<br />
US<br />
3. RT US<br />
4. RT ST US ST<br />
5. RS TU<br />
6. ___<br />
RS ___<br />
TU<br />
7. R U<br />
8. VST WTS<br />
9. mVST mWTS<br />
10. VSR is <strong>the</strong> complement of VST.<br />
11. WTU is <strong>the</strong> complement of WTS.<br />
12. VSR WTU<br />
13. RVS UWT (ASA ASA)<br />
14 1. ____<br />
MQ ____<br />
NQ<br />
2. ___<br />
QP ____<br />
QO<br />
3. ___<br />
PQ ____<br />
MQ<br />
4. MQP is a right angle.<br />
5. ____<br />
OQ ____<br />
NQ<br />
6. NQO is a right angle.<br />
7. MQP NQO (Right angles are<br />
congruent.)<br />
8. PQO PQO<br />
9. MQP PQO NQO PQO<br />
10. MQO NQP<br />
11. MQO NQP (SAS SAS)<br />
15 1. ___<br />
AC ___<br />
BC<br />
2. ACF BCG<br />
3. DCF ECG<br />
4. DCF ACF BCG ACF<br />
5. DCF ACF BCG ECG<br />
6. ACD BCE<br />
7. CAF CBA<br />
8. CAD CBE (ASA ASA)<br />
9. ___<br />
DC ___<br />
EC<br />
10. DCE is isosceles. ( Definition of an<br />
isosceles triangle)<br />
16 Draw a line longer than <strong>the</strong> sum of <strong>the</strong><br />
lengths of <strong>the</strong> two segments. Copy ___<br />
AB onto<br />
<strong>the</strong> new line. Place <strong>the</strong> compass vertex where<br />
<strong>the</strong> arc swing intersects <strong>the</strong> line and mark off<br />
<strong>the</strong> length of ___<br />
CD . The line segment from <strong>the</strong><br />
original vertex to <strong>the</strong> final arc swing marks<br />
off <strong>the</strong> new segment.<br />
17 Bisect ___<br />
AB and <strong>the</strong>n bisect each half of <strong>the</strong><br />
original segment.<br />
18 Use angle bisector procedure.<br />
19 Bisect side ___<br />
AB . Mark <strong>the</strong> point where <strong>the</strong><br />
bisector intersects <strong>the</strong> line M. Draw a line<br />
from C to M.<br />
Chapter Review 27
20 Use constructing congruent angles procedure.<br />
21 Use angle bisector procedure on each side of<br />
<strong>the</strong> triangle. The angle bisectors should meet<br />
at a common point, P.<br />
22 Use constructing a perpendicular bisector<br />
procedure.<br />
Trans<strong>for</strong>mations and<br />
<strong>the</strong> Coordinate Plane<br />
6-1 Cartesian Coordinate<br />
System<br />
(pages 91–92)<br />
1 (2) 5<br />
2 (2) 6<br />
3 (4) 10<br />
4 (2) 6<br />
5 (1) √ 2<br />
6 (2) √ 226<br />
7 a A(5, 2), B(3, 3) b A(3, 4), B(0, 5)<br />
c A(x, y), B(0, 0)<br />
8 The length of base ___<br />
AB is <strong>the</strong> difference<br />
between <strong>the</strong> x-coordinates of A and B;<br />
6 2 4. The height is <strong>the</strong> vertical distance<br />
from C to ___<br />
AB or <strong>the</strong> difference between <strong>the</strong><br />
y-coordinates; 4 1 3. There<strong>for</strong>e,<br />
A 1 _ bh <br />
2 1 _ (4)(3) 6.<br />
2<br />
9 The length of <strong>the</strong> base ___<br />
AB is <strong>the</strong> difference<br />
between <strong>the</strong> x-coordinates of A and B;<br />
1 (3) 4. The height is <strong>the</strong> vertical distance<br />
from C to ___<br />
AB or <strong>the</strong> difference between<br />
<strong>the</strong> y-coordinates; 8 4 4. There<strong>for</strong>e,<br />
A 1 _ bh <br />
2 1 _ (4)(4) 8.<br />
2<br />
28 Chapter 6: Trans<strong>for</strong>mations and <strong>the</strong> Coordinate Plane<br />
23 Copy one side and <strong>the</strong> angles at each vertex.<br />
Extend <strong>the</strong> rays until <strong>the</strong>y meet.<br />
24 Use <strong>the</strong> line bisector procedure. Mark <strong>the</strong><br />
intersection of <strong>the</strong> line and its bisector as <strong>the</strong><br />
midpoint.<br />
CHAPTER<br />
6<br />
10 The length of <strong>the</strong> base −−<br />
AB is <strong>the</strong> difference in<br />
<strong>the</strong> y-coordinates of A and B; 6 2 4. The<br />
height is <strong>the</strong> horizontal distance from C<br />
to −−<br />
AB or <strong>the</strong> difference between <strong>the</strong><br />
x-coordinates; 3 2 1. There<strong>for</strong>e,<br />
A 1 _ bh <br />
2 1 _ (4)(1) 2.<br />
2<br />
11 The length of <strong>the</strong> base ___<br />
AB is <strong>the</strong> difference<br />
between <strong>the</strong> y-coordinates of A and B;<br />
8 2 6. The height is <strong>the</strong> horizontal distance<br />
from C to ___<br />
AB or <strong>the</strong> difference between<br />
<strong>the</strong> x-coordinates; 4 (2) 6. There<strong>for</strong>e,<br />
A 1 _ bh <br />
2 1 _ (6)(6) 18.<br />
2<br />
12 D(x, y) (1, 2)<br />
13 Draw a horizontal line from A to C. For<br />
ABC, <strong>the</strong> length of <strong>the</strong> base ___<br />
AC is <strong>the</strong> difference<br />
between <strong>the</strong> x-coordinates of A and C;<br />
3 (7) 10. The height is <strong>the</strong> vertical distance<br />
from B to ___<br />
AC or <strong>the</strong> difference between<br />
<strong>the</strong> y-coordinates; 5 1 4. The area of<br />
ABC is A 1 _ bh <br />
2 1 _ (10)(4) 20. For<br />
2<br />
ADC, <strong>the</strong> base is 10 and <strong>the</strong> height is 4. The<br />
area of ADC is A 1 _ bh <br />
2 1 _ (10)(4) 20.<br />
2<br />
The area of quadrilateral ABCD is 20 20 <br />
40.
14 The distance from A to B is<br />
√ <br />
[2 (7) ] 2 (5 1 ) 2 <br />
5 2 4 2 41 . The distance from B to C<br />
is √ <br />
[3 (2) ] 2 (1 5 ) 2 <br />
5 2 4 2 41 . The distance from C to D<br />
is<br />
√ <br />
[(2) 3 ] 2 [(3) 1 ] 2 <br />
5 2 4 2 41 . The distance from D to A<br />
is √ <br />
[(7) (2)] 2 [(1 (3)] 2<br />
5 2 4 2 41 . The perimeter of ABCD<br />
41 41 41 41 4 41 .<br />
15 The length of <strong>the</strong> base ___<br />
AC is <strong>the</strong> difference<br />
between <strong>the</strong> x-coordinates of A and C;<br />
6 (4) 10. The height is <strong>the</strong> vertical distance<br />
from B to ___<br />
AC or <strong>the</strong> difference between<br />
<strong>the</strong> y-coordinates; 6 (6) 12. There<strong>for</strong>e,<br />
A 1 _ bh <br />
2 1 _ (10)(12) 60.<br />
2<br />
16 Using <strong>the</strong> distance <strong>for</strong>mula, AB BC 13<br />
and CA 10. The perimeter of ABC <br />
13 13 10 36.<br />
17 Isosceles because PA AT 5<br />
18 Scalene because AB √ 53 , CB √ 74 , and<br />
AC √ 29<br />
19 Scalene because MA √ 130 , AD √ 26 ,<br />
and MD √ 104<br />
20 Scalene because WI √ 26 , IT √ 125 , and<br />
WT √ 109<br />
21 All three sides measure 2, <strong>the</strong>re<strong>for</strong>e <strong>the</strong> triangle<br />
must be equilateral.<br />
22 The radius is <strong>the</strong> distance from <strong>the</strong> center to<br />
any point on <strong>the</strong> circle.<br />
r √ <br />
(4 1 ) 2 (2 2 ) 2 5<br />
Diameter 2(5) 10<br />
23 The two diagonals are congruent because<br />
AD CB 2.<br />
24 A B 2 [(1) 3 ] 2 (2 2 ) 2 32<br />
B C 2 (3 1 ) 2 (2 4 ) 2 8<br />
A C 2 [(1) 1 ] 2 [(2) 4 ] 2 40.<br />
32 8 40 or A B 2 B C 2 A C 2 .<br />
6-2 Translations<br />
(pages 94–95)<br />
1 (1) (1, 2)<br />
2 (2) (3, 2)<br />
3 (4) T 8, 4<br />
4 (4) 6<br />
5 T 0, 2<br />
6 T 3, 0<br />
7 (4, 1)<br />
8 (3, 7)<br />
9 (2, 4)<br />
10 (4, 3)<br />
11 (4, 2)<br />
12 (2, 2)<br />
13 T 3, 2<br />
14 T 2, 2<br />
15 (3, 0)<br />
16 (7, 3)<br />
17 (4, 7)<br />
18 K(8, 5) → (2, 9), E(10, 3) → (4, 1),<br />
N(2, 2) → (8, 6)<br />
19<br />
y<br />
20<br />
D<br />
10<br />
9<br />
8<br />
7<br />
D"<br />
6<br />
5<br />
4<br />
3 W"<br />
D'<br />
2<br />
1<br />
W<br />
5 4 3 2 1<br />
1<br />
2<br />
3<br />
4<br />
5<br />
1 2 3 4<br />
W'<br />
5 6 7 8 9 10<br />
E<br />
E'<br />
E"<br />
a D’(1, 2), E’(6, 3), W’(2, 1)<br />
b D(0, 6), E(7, 7), W(3, 3)<br />
c T 3, 1<br />
A'<br />
W'<br />
y<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
5 4 3 2 1 1 2 3 4 5 6 7 8 9 10<br />
1<br />
2<br />
3<br />
4<br />
5<br />
S'<br />
S(3, 4)<br />
H'<br />
A"<br />
A(0, 1)<br />
W"<br />
W(1, 2)<br />
S"<br />
H(5, 1)<br />
a W’(2, 0), A’(3, 3), S’(0, 6), H’(2, 3)<br />
b W(5, 1), A(4, 2), S(7, 5), H(9, 2)<br />
c T 4 , 1<br />
H"<br />
x<br />
x<br />
6-2 Translations 29
6-3 Line Reflections and<br />
Symmetry<br />
(pages 100–101)<br />
1 (3) A<br />
2 (3) V<br />
3 (1) only vertical line symmetry<br />
4 (2) 2<br />
5 (3) (4, 7)<br />
6 y-axis<br />
7 both<br />
8 x-axis<br />
9 y-axis<br />
10 P’(1, 8)<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
2<br />
3<br />
4<br />
y<br />
5<br />
6<br />
7<br />
8<br />
9<br />
10<br />
P(2, 4)<br />
6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10<br />
1<br />
11 P’(3, 7)<br />
P(3, 3)<br />
P'(3, 7)<br />
P'(1, 8)<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
y 2<br />
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8<br />
1<br />
12 P’(4, 1)<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
y 2<br />
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
P'(4, 1)<br />
y 2<br />
P(4, 5)<br />
30 Chapter 6: Trans<strong>for</strong>mations and <strong>the</strong> Coordinate Plane<br />
x<br />
x<br />
x<br />
13 (5, 2)<br />
14 (2, 3)<br />
15 (2, 5)<br />
16 A’(2, 1), B’(3, 4), C’(4, 5)<br />
C(4, 5)<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8<br />
1<br />
A'(2, 1)<br />
2<br />
C'(4, 5)<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
B(3, 4)<br />
A(2, 1)<br />
B'(3, 4)<br />
17 G’(0, 0), H’(4, 3), I’(1, 4), J’(5, 2)<br />
5<br />
I'(–1, 4)<br />
4<br />
H'(–4, 3)<br />
3<br />
I(1, 4)<br />
J(–5, 2)<br />
2<br />
1<br />
6<br />
y<br />
H(4, 3)<br />
J'(5, 2)<br />
6 5 4 3 21G<br />
G'1<br />
2 3 4 5 6<br />
1<br />
2<br />
18 A’(3, 3), B’(5, 8), C’(1, 5)<br />
y<br />
10<br />
9<br />
8<br />
7<br />
6<br />
B'(5, 8)<br />
C'(1, 5 5)<br />
4<br />
3A'(3,<br />
3)<br />
B(8, 5)<br />
2<br />
1<br />
6 5 4 3 2 1<br />
1<br />
2<br />
3<br />
4<br />
A(3, 3)<br />
1 2 3 4<br />
C(5, 1)<br />
5 6 7 8 9 10<br />
x<br />
y x<br />
19 Q’(2, 4), Z’(3, 4), D’(1, 2)<br />
5<br />
Z'(3, 4)<br />
4<br />
Z(4, 3)<br />
3<br />
Q'(2, 4)<br />
2<br />
1<br />
D'(1, 2)<br />
6<br />
y<br />
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8<br />
1<br />
D<br />
Q(4, 2) (2, 1)<br />
3<br />
4<br />
y x<br />
x<br />
x<br />
x
20<br />
y x 2<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
y<br />
y x<br />
65 4 3 2 1<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
1 2 3 4 5 6<br />
x y 2<br />
6-4 Point Reflection and<br />
Symmetry<br />
(pages 103–104)<br />
1 (2) H<br />
2 (1) only line symmetry<br />
3 (3) both point and line symmetry<br />
4 (3) (k, 2k)<br />
5 line symmetry<br />
6 both<br />
7 point symmetry<br />
8 both<br />
9 (1, 3)<br />
10 (3, 1)<br />
11 (9, 0)<br />
12 (2, 6)<br />
13 (4, 4)<br />
14 (8, 2)<br />
15 (2, 4)<br />
16 (2, 8)<br />
17 (10, 10)<br />
18 Point reflection is a trans<strong>for</strong>mation of a<br />
figure into ano<strong>the</strong>r figure.<br />
Point symmetry is a quality of a figure. A figure<br />
with point symmetry is not changed by a<br />
reflection through that point. (It is rotated in<br />
ei<strong>the</strong>r direction 180 about a point.)<br />
x<br />
6-5 Rotations<br />
(page 107)<br />
1 (2) trapezoid<br />
2 (1) regular hexagon<br />
3 (2) regular pentagon<br />
4 (2) rectangle<br />
5 (1) equilateral triangle<br />
6 (1)<br />
7 Z<br />
8 X<br />
9 Y<br />
10 Z<br />
11 (1, 5)<br />
12 (3, 3)<br />
13 (2, 8)<br />
14 (2, 3)<br />
15 (2, 2)<br />
16 (4, 4)<br />
17 (6, 3)<br />
18 (2, 6)<br />
19 (5, 5)<br />
20 W(3, 2), H(8, 2), Y(5, 10)<br />
6-6 Dilations<br />
(page 110)<br />
1 (3) (4, 10)<br />
2 (2) 1 _<br />
2<br />
3 (4) (x, y) → (2x, 2y)<br />
4 (4) 20 square inches<br />
5 (6, 9)<br />
9<br />
6 ( 3, _<br />
2 ) or (3, 4.5)<br />
7 (4, 6)<br />
For problems 8–11, see figure below.<br />
8 A(0, 8), B(2, 2), C(6, 4)<br />
9 A(0, 3), B ( 3 _ , <br />
4 3 _<br />
4 ) , C ( 2<br />
1 _ , 1<br />
4 1 _<br />
2 )<br />
10 A(0, 12), B(3, 3), C(9, 6)<br />
6-6 Dilations 31
11 A(0, 8), B(2, 2), C(6, 4)<br />
(8)<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
A<br />
4<br />
3<br />
C<br />
B<br />
2<br />
1<br />
109 8 7 6 5 4 3 2 1<br />
1<br />
2<br />
3<br />
1 2<br />
(9)<br />
3 4 5 6 7 8 9 10<br />
4<br />
5<br />
6<br />
(11)<br />
7<br />
8<br />
9<br />
10<br />
11<br />
12<br />
13<br />
14<br />
(10)<br />
y<br />
12 (1, √ 3 )<br />
13 ( √ 2 _ ,<br />
2 √ 2 _<br />
2 )<br />
14 (3a, 3b)<br />
15 k 1 _ ; (2, 4) → (1, 2)<br />
3<br />
16 k 1 _ ; (2, 4) → (1, 2)<br />
2<br />
17 (3, 12)<br />
18 G(1, 1), N(4, 1), A(4, 1), T(1, 1)<br />
6-7 Properties Under<br />
Trans<strong>for</strong>mations<br />
(pages 111–113)<br />
1 (2) dilation<br />
2 (4) dilation<br />
3 (2) translation<br />
4 (2) r y x<br />
5 (3) dilation D 1<br />
6 r y-axis<br />
7 T 5, 0<br />
8 R 270 or R 90<br />
9 T 4, 3<br />
32 Chapter 6: Trans<strong>for</strong>mations and <strong>the</strong> Coordinate Plane<br />
x<br />
10<br />
11<br />
12<br />
13<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
y<br />
1<br />
x<br />
8 7 6 5 4 3 2 1<br />
1<br />
K'(2, 2)<br />
1 2 3 4<br />
K(2, 2)<br />
5 6 7 8<br />
M'(5, 2)<br />
2<br />
3<br />
M(5, 2)<br />
4<br />
Z'(3, 4)<br />
5<br />
6<br />
7<br />
8<br />
Z(3, 4)<br />
direct isometry<br />
y<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
K'(1, 1) 2<br />
1<br />
M'(2, 1)<br />
x<br />
8 7 6 5 4 3 2 1 1 2 3 4<br />
1 Z'(0, 1)<br />
5 6 7 8<br />
2 K(2, 2)<br />
3<br />
M(5, 2)<br />
4<br />
5<br />
6<br />
7<br />
8<br />
Z(3, 4)<br />
direct isometry<br />
5<br />
4<br />
3<br />
2<br />
y<br />
1<br />
x<br />
5 4 3 2 1<br />
1<br />
1 2 3 4 5 6 7 8 9 10 11 12<br />
K(2, 2 2) M(5, 2)<br />
3<br />
4<br />
5<br />
K'(4, 4)<br />
Z(3, 4)<br />
W'(10, 4)<br />
6<br />
7<br />
8<br />
9<br />
10<br />
not an isometry<br />
A'''<br />
R"<br />
A" R'''<br />
C'''<br />
C"<br />
Z'(6, 8)<br />
y<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
109 8 7 6 5 4 3 2 1<br />
1<br />
C<br />
2<br />
1 2 3 4<br />
R<br />
5 6 7 8 9 10<br />
3<br />
4<br />
A<br />
5<br />
6<br />
C'<br />
R'<br />
7<br />
8<br />
9<br />
10<br />
A'<br />
a C(3, 5), A(4, 7), R(7, 4)<br />
x
14<br />
15<br />
b C(3, 5), A(4, 7), R(7, 4)<br />
c C(5, 3), A(7, 4), R(4, 7)<br />
d c<br />
D"<br />
D'''<br />
I"<br />
K"<br />
2<br />
1<br />
K'<br />
11109 8 7 6 5 4 3 2 1<br />
1<br />
I'''<br />
2<br />
K'''<br />
3<br />
4<br />
5<br />
1 2 3 4 5 6 7 8<br />
y<br />
12<br />
11<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
I<br />
K<br />
D<br />
I'<br />
D'<br />
9 10 11<br />
a K(6, 1), I(9, 2), D(10, 7)<br />
b K(6, 1), I(9, 2), D(10, 7)<br />
c K(1, 2), I(4, 1), D(5, 4)<br />
10<br />
9<br />
8<br />
7<br />
6<br />
A"<br />
R'''<br />
5<br />
4<br />
3<br />
R"<br />
A'''<br />
2<br />
1<br />
J'''<br />
J"<br />
J' J<br />
109 8 7 6 5 4 3 2 1<br />
1<br />
2<br />
3<br />
1 2 3 4<br />
A'<br />
R'<br />
5 6 7<br />
A<br />
8 9 10<br />
4<br />
5<br />
R<br />
a J(1, 0), A(3, 1), R(2, 2)<br />
b J(0, 2), A(2, 6), R(4, 4)<br />
c J(2, 0), A(6, 2), R(4, 4)<br />
6-8 Composition of<br />
Trans<strong>for</strong>mations<br />
(pages 117–119)<br />
1 (3) R 200<br />
2 (2) ___<br />
AT<br />
3 (1) (x, y)<br />
4 (3) r y x D 3<br />
5 (1) a direct isometry<br />
6 (2) (x, y)<br />
7 (4) (3, 8)<br />
8 (4) D<br />
9 (3) T 4, 0<br />
10 (1) (x, y)<br />
11 (3) orientation<br />
12 N<br />
y<br />
x<br />
x<br />
13 A<br />
14 R 90<br />
15 Any combination of three rotations, <strong>the</strong> sum<br />
of whose angles is 100<br />
16 Any combination of two rotations, <strong>the</strong> sum<br />
of whose angles is 180<br />
17 Glide reflection<br />
18 (6, 1)<br />
19 (2, 3)<br />
20 (2, 7)<br />
21<br />
y<br />
22<br />
10<br />
9<br />
8<br />
7<br />
P'<br />
M<br />
y x<br />
6<br />
5<br />
M'<br />
4<br />
3<br />
P<br />
2<br />
C<br />
1<br />
C'<br />
109 8 7 6 5 4 3 2 1<br />
1<br />
C"<br />
2<br />
3<br />
4<br />
1 2 3 4 5 6 7 8 9 10<br />
M"<br />
5<br />
6<br />
7<br />
P"<br />
8<br />
9<br />
10<br />
C(2, 1), M(7, 5), P(4, 8)<br />
C(2, 1), M(7, 5), P(4, 8)<br />
d r y x<br />
Y'<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
Y<br />
X"<br />
X<br />
4<br />
3<br />
2<br />
X'<br />
Z'<br />
1<br />
Z<br />
109 8 7 6 5 4 3 2 1<br />
1<br />
1 2 3 4 5 6 7 8 9 10<br />
2<br />
3<br />
4<br />
5<br />
Y"<br />
y x<br />
6<br />
7<br />
8<br />
9<br />
10<br />
Z"<br />
X(5, 3), Y(2, 6), Z(7, 1)<br />
X(3, 5), Y(6, 2), Z(1, 7)<br />
d (1) rotation<br />
Chapter Review (pages 119–121)<br />
1 (1) I<br />
2 (1) WOW<br />
3 (3) parallel to <strong>the</strong> y-axis<br />
4 (3) (5, 2)<br />
y<br />
Chapter Review 33<br />
x<br />
x
5 (3) (x, y) → (x, 2y)<br />
6 (1) translation<br />
7 (1) rotation<br />
8 (1) A<br />
9 (4) D<br />
10 (2) (5, 4)<br />
11 (3) reflection in <strong>the</strong> line y x<br />
12 (2) (x, y) → (4x, 2y)<br />
13 (4) y 0<br />
14 (3) r x 1 r y-axis<br />
15 (1) _ 1<br />
2<br />
16 (4)<br />
17 (3) (6, 1)<br />
18 5<br />
19 13<br />
20 2 √ 10<br />
21 13<br />
22 3 √ 10<br />
23 (1, 1)<br />
24 (2, 6)<br />
25 (3, 0)<br />
26 (4, 1)<br />
27 r y x (1, 8) (8, 1)<br />
Polygon Sides and<br />
Angles<br />
7-1 Basic Inequality<br />
Postulates<br />
(pages 125–126)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
34 Chapter 7: Polygon Sides and Angles<br />
28 (4, 9)<br />
29 a (2, 2)<br />
b (4, 0)<br />
c (2, 2)<br />
d (2, 2)<br />
e (3, 2)<br />
30 a and c<br />
H<br />
C<br />
10<br />
8<br />
6<br />
4<br />
2<br />
14 12 10 8 6 4 2<br />
2 4 6 8 10 12<br />
T"(8, 2)<br />
14<br />
2<br />
b r y x<br />
C"(2, 10)<br />
4<br />
6<br />
8<br />
10<br />
y<br />
T<br />
C'(4, 3)<br />
CHAPTER<br />
7<br />
T'(9, 1)<br />
H'(10, 3)<br />
in each problem. (These solutions are intended<br />
to be used as a guide— o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. m1 m2<br />
2. m2 m1<br />
3. m2 m3<br />
4. m3 m2<br />
5. m3 m1 ( Transitive postulate of<br />
inequality)<br />
6. m1 m3<br />
H"(10, 10)<br />
x
2 1. PQ PS<br />
2. ___<br />
PQ −−<br />
PS<br />
3. 1 2<br />
4. m1 m2<br />
5. m1 m3<br />
(Isosceles triangle<br />
<strong>the</strong>orem)<br />
6. m2 m3<br />
3 1. CE BD<br />
2. CF BF<br />
3. CE CF FE<br />
4. CF FE BD<br />
5. BD BF FD<br />
(Substitution<br />
postulate of<br />
inequality)<br />
6. CF FE BF FD<br />
7. BF FE BF FD<br />
8. BF FE BF (Subtraction<br />
BF FD BF postulate of<br />
or FE FD<br />
4 1.<br />
inequality)<br />
___<br />
AE ___<br />
EB<br />
2. ABE BAE<br />
3. mABE mBAE<br />
4. mDAB mCBA<br />
(Isosceles triangle<br />
<strong>the</strong>orem)<br />
5. mDAB mCAD mBAE<br />
6. mCAD mBAE mCBA<br />
7. mCBA mABE mDBC<br />
8. mCAD mBAE<br />
mABE mDBC<br />
9. mCAD mABE<br />
mABE mDBC<br />
10. mCAD mABE (Subtraction<br />
mABE mABE postulate of<br />
mDBC mABE<br />
or mCAD mDBC<br />
5 1. C is <strong>the</strong> midpoint of<br />
inequality)<br />
−−<br />
AB .<br />
2. AC CB<br />
3. AB AC CB<br />
4. AB AC AC<br />
or AB 2AC<br />
5. AB _ AC<br />
2<br />
6. F is <strong>the</strong> midpoint of −−<br />
DE .<br />
7. DF DF<br />
8. DE DF FE<br />
9. DE DF DF<br />
or DE 2DF<br />
10. DE _ DF<br />
2<br />
11. AC DF<br />
12. AB _ DF<br />
2<br />
13. AB _ <br />
DE<br />
2 _<br />
2<br />
14. AB _ 2 <br />
DE<br />
2 _<br />
2 2 (Multiplication<br />
or AB DE postulate of<br />
equality.)<br />
6 1. RP 3RS<br />
2. RP _ 3RS<br />
<br />
3 _<br />
3<br />
or RP _ RS<br />
3<br />
3. RQ 3RT<br />
4. RQ _ 3RT<br />
<br />
3 _<br />
3<br />
or RQ _ RT<br />
3<br />
5. RS RT<br />
6. RP _ RT<br />
3<br />
7. RP _ RQ<br />
<br />
3 _<br />
3<br />
8. RP _ RQ<br />
3 <br />
3 _ 3 (Multiplication<br />
3<br />
or RP RQ postulate of<br />
equality.)<br />
7 1. I is <strong>the</strong> midpoint of −−<br />
EH .<br />
2. EH EI IH<br />
3. EI IH<br />
4. EH EI EI<br />
EH 2EI<br />
5. J is <strong>the</strong> midpoint of −−<br />
EF .<br />
6. EF EJ JF<br />
7. EJ JF<br />
8. EF EJ EJ<br />
or EF 2EJ<br />
9. EH EF<br />
10. 2EI EF<br />
11. 2EI 2EJ<br />
12. EI EJ (Division postulate<br />
of equality)<br />
8 1. ___<br />
BD bisects ABC.<br />
2. ABD DBC (Definition of<br />
bisector)<br />
3. mABD mDBC<br />
4. mABC mABD mDBC<br />
5. mABC mABD mABD<br />
or mABC 2mABD<br />
6. −−<br />
BD bisects ADC.<br />
7. ADB BDC<br />
8. mADB mBDC<br />
9. mADC mADB mBDC<br />
10. mADC mADB mADB<br />
or mADC 2mADB<br />
7-1 Basic Inequality Postulates 35
11. mABC mADC<br />
12. 2mABD 2mADB<br />
13. mABD mADB<br />
(Division postulate of equality)<br />
9 1. AE 1 _ AC<br />
3<br />
2. AE 3 1 _<br />
AC 3 or 3AE AC<br />
3<br />
(Multiplication postulate of equality)<br />
3. AD 1 _ AB<br />
3<br />
4. AD 3 1 _<br />
AB 3<br />
3<br />
or 3AD AB<br />
5. AC AB<br />
6. 3AE 3AD (Substitution postulate)<br />
7. AE AD (Division postulate of<br />
inequality)<br />
10 1. ___<br />
AE is <strong>the</strong> median from A to ___<br />
BC .<br />
2. E is <strong>the</strong> midpoint of ___<br />
BC .<br />
(Definition of median)<br />
3. BE EC (Definition of midpoint)<br />
4. BC BE EC<br />
5. BC BE BE<br />
or BC 2BE<br />
6. ___<br />
CD is <strong>the</strong> median from C to ___<br />
AB .<br />
7. D is <strong>the</strong> midpoint of ___<br />
AB .<br />
(Definition of median)<br />
8. AD DB<br />
9. AB AD DB<br />
10. AB AD AD<br />
or AB 2AD<br />
11. AB BC<br />
12. 2AD 2BE<br />
13. AD BE (Division postulate of<br />
inequality)<br />
7-2 The Triangle Inequality<br />
Theorem<br />
(page 128)<br />
1 Yes<br />
2 No<br />
3 Yes<br />
4 No<br />
5 Yes<br />
6 No<br />
36 Chapter 7: Polygon Sides and Angles<br />
7 Yes<br />
8 Yes<br />
9 Yes<br />
10 Yes<br />
11 1 S 9<br />
12 6 S 10<br />
13 2.5 S 5.5<br />
14 0 S 12<br />
15 1 1 _ S 3<br />
2 1 _<br />
2<br />
7-3 The Exterior Angles<br />
of a Triangle<br />
(pages 130–132)<br />
1 (2) isosceles<br />
2 (4) right<br />
3 a CAD<br />
b ABC and ACB<br />
c mCAD mABC; mCAD mACB<br />
4 a True<br />
b True<br />
c True<br />
d True<br />
5 6 and 9<br />
6 5, 9, 8, and 10<br />
7 7 and 10<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
8 1. m1 m3 (Exterior angle<br />
<strong>the</strong>orem)<br />
2. m3 m5 (Exterior angle<br />
<strong>the</strong>orem)<br />
3. m1 m5 (Transitive postulate<br />
of inequality)<br />
9 1. ___<br />
AB ___<br />
CD<br />
2. 1 is a right angle.<br />
3. AED is a right angle.<br />
4. AED 1 (Right angles are<br />
congruent.)
5. mAED m1<br />
6. mAED m2 (Exterior angle<br />
<strong>the</strong>orem)<br />
7. m1 m2<br />
10 1. mDCB mCBA (Exterior angle<br />
<strong>the</strong>orem)<br />
2. mCBA mCBM mMBA<br />
3. mCBA mMBA<br />
4. mDCB mMBA (Transitive postulate<br />
of inequality)<br />
11 12x 8 (4x 6) (7x 6)<br />
x 8<br />
12 12x 3 2 (5x 5) (5x 5)<br />
x 11<br />
13 Let x be mB, <strong>the</strong>n mA is 3x.<br />
3x x 104<br />
x 26<br />
mA 3x 3(26) 78<br />
14 If <strong>the</strong> vertex angle of <strong>the</strong> triangle measures<br />
130, each base angle measures 25 and<br />
each exterior angle at <strong>the</strong> base measures<br />
180 25 155.<br />
7-4 Inequalities Involving<br />
Sides and Angles of<br />
Triangles<br />
(page 133)<br />
1 ZXY<br />
2 PRQ<br />
3 ___<br />
AC<br />
4 GFH<br />
5 ___<br />
KL<br />
6 ACB<br />
7 ___<br />
YZ<br />
8 mDGE mDEF mGDE, so<br />
mDGE mGDE and DE EG.<br />
9 PQS QSP, but QSP SQR, so<br />
PQS SQR<br />
10 m4 m3 m1, so m4 m1<br />
Chapter Review (pages 133–136)<br />
1 (3) p m p n<br />
2 (3) 28<br />
3 (2) an obtuse angle<br />
4 True. Transitive postulate of inequality<br />
5 True. Additive postulate of inequality<br />
6 True. Postulate of inequality<br />
7 False<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable statements<br />
to complete each proof. Depending on <strong>the</strong><br />
textbook used, <strong>the</strong> wording and <strong>for</strong>mat of reasons<br />
may differ, so <strong>the</strong>y have not been supplied <strong>for</strong> <strong>the</strong><br />
method of congruence applied in each problem.<br />
(These solutions are intended to be used as a<br />
guide—o<strong>the</strong>r possible solutions may vary.)<br />
8 1. m2 m3<br />
2. m1 m2 (Exterior angle<br />
<strong>the</strong>orem)<br />
3. m1 m3 (Transitive postulate<br />
of inequality)<br />
9 1. BEST is a parallelogram.<br />
2. BT BE<br />
3. ___<br />
BT ___<br />
ES (Definition of a<br />
parallelogram)<br />
4. BT ES<br />
5. ES BE (Substitution<br />
postulate of<br />
inequality)<br />
10 1. ___<br />
GS ___<br />
GD<br />
2. NPS RSP (Isosceles triangle<br />
<strong>the</strong>orem)<br />
3. mNPS mRSP<br />
4. mISR mIPN<br />
5. mISR mRSP mIPN mRSP<br />
(Addition postulate<br />
of inequality)<br />
6. mISP mRSP mIPN mNPS<br />
7. mISP mIPS (Postulate of<br />
inequality)<br />
11 1. mBCE mDBC<br />
2. mACE mABD<br />
3. mBCE mACE (Addition<br />
mDBC mABD postulate<br />
inequality)<br />
4. mACB mABC (Postulate of<br />
inequality)<br />
12 1. ___<br />
AD ___<br />
DC<br />
2. ACD CAD (Isosceles triangle<br />
<strong>the</strong>orem)<br />
3. mACD mCAD<br />
4. mDAB mDCB<br />
5. mDAB mACD ( Subtraction<br />
mDCB mACD postulate of<br />
inequality)<br />
Chapter Review 37
6. mDAB mCAD <br />
mDCB mACD<br />
7. mCAB mACB (Postulate of<br />
inequality)<br />
13 1. PQRS is a parallelogram.<br />
2. SP RQ (Definition of<br />
parallelogram)<br />
3. ST TP RQ<br />
4. ST TP QU RU<br />
or ST RU QU TP<br />
5. TP QU<br />
6. 0 QU TP (Subtraction<br />
postulate of<br />
inequality)<br />
7. 0 ST RU<br />
8. ST RU (Addition<br />
postulate of<br />
inequality)<br />
14 ___<br />
ST<br />
15 ___<br />
AB<br />
16 ___<br />
AB<br />
17 ___<br />
BC<br />
18 −−<br />
DE<br />
19 AD BD<br />
20 mB 120<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
21 1. ___<br />
AD bisects CAB.<br />
2. ___<br />
BE bisects CBA.<br />
3. mDAB mEBA<br />
4. mDAB 2 mEBA 2 (Multiplication<br />
postulate of<br />
inequality)<br />
5. mCAB 2(mDAB) (Definition<br />
of bisector)<br />
6. mCBA 2(mEBA) (Definition<br />
of bisector)<br />
7. mCAB mCBA<br />
38 Chapter 7: Polygon Sides and Angles<br />
22 1. CF AE<br />
2. CF 2 AE 2 (Multiplication postlate<br />
of inequality)<br />
3. F is <strong>the</strong> midpoint of ___<br />
CD .<br />
4. CF 2 CD (Definition of<br />
midpoint)<br />
5. CD AE 2<br />
6. E is <strong>the</strong> midpoint of ___<br />
AB .<br />
7. AB AE 2 (Definition of<br />
midpoint)<br />
8. CD AB (Multiplication postulate<br />
of inequality)<br />
23 1. AC CB<br />
2. AC _<br />
2<br />
CB<br />
_<br />
2<br />
3. AD AC _<br />
2<br />
4. AD CB _<br />
2<br />
5. BE CB _<br />
2<br />
(Division postulate of<br />
inequality)<br />
(Definition of<br />
midpoint)<br />
(Definition of<br />
midpoint)<br />
6. AD BE<br />
24 Let A be <strong>the</strong> acute angle and let B be<br />
its supplement. mB 180 mA. Since<br />
mA 90, mB 90 and is <strong>the</strong>re<strong>for</strong>e<br />
obtuse by <strong>the</strong> subtraction postulate of<br />
inequality.<br />
25 Let C be <strong>the</strong> complement of A, and<br />
let D be <strong>the</strong> complement of B. mC <br />
90 mA and mC mD by <strong>the</strong> subtraction<br />
postulate of inequality.<br />
26 m1 m3 because an exterior angle is<br />
greater than ei<strong>the</strong>r nonadjacent interior angle<br />
and m1 m2. So m2 m3 by <strong>the</strong><br />
substitution postulate of inequality.<br />
27 mABC mABD because a whole is<br />
greater than its parts. mBAC mABC<br />
because <strong>the</strong>y are opposite congruent sides<br />
of a triangle. So mBAC mABD by <strong>the</strong><br />
substitution postulate of inequality, and<br />
DB DA because <strong>the</strong> greater side lies<br />
opposite <strong>the</strong> greater angle.
28 mBDA mACB because a whole is<br />
greater than its parts. mACB mBCA<br />
because <strong>the</strong>y are opposite congruent sides<br />
of a triangle. So mBDA mBCA by <strong>the</strong><br />
substitution postulate of inequality, and<br />
AB AD because <strong>the</strong> greater side lies<br />
opposite <strong>the</strong> greater angle.<br />
29 Draw diagonal −−<br />
AC , <strong>for</strong>ming two triangles.<br />
In ABC, mBCA mBAC because<br />
<strong>the</strong>y are opposite congruent sides. In<br />
ADC, mACD mCAD because <strong>the</strong><br />
greater angle is opposite <strong>the</strong> greater side.<br />
mBCA mACD mBAC mCAD<br />
or mBCD mBAD by <strong>the</strong> addition postulate<br />
of inequality.<br />
30 Look at <strong>the</strong> two right triangles <strong>for</strong>med with<br />
common side −−<br />
EG . mD mF by <strong>the</strong> subtraction<br />
postulate of inequality, so DE EF<br />
because <strong>the</strong> greater side lies opposite <strong>the</strong><br />
greater angle.<br />
CHAPTER<br />
8<br />
8-1 The Slope of a Line<br />
(pages 141–144)<br />
1 (2) (0, 4)<br />
2 (2) It has an x-intercept of 2.<br />
3 (4) y 3<br />
4 (3) a y-intercept of 5<br />
5 (3)<br />
6 (2) x 1<br />
7 m 5 _ ; negative<br />
3<br />
8 m 2; positive<br />
9 m 3 _ ; positive<br />
4<br />
31 m2 180 114 66<br />
m1 180 50 66 64<br />
32 x 4 5 105<br />
x 60<br />
33 y 18 0 110 70<br />
x 7 0 130<br />
x 60<br />
34 True<br />
35 False<br />
36 True<br />
37 False<br />
38 False<br />
Slopes and Equations<br />
of Lines<br />
10 Undefined<br />
11 m 2; negative<br />
12 m 3 _ ; positive<br />
2<br />
13 m 0; zero<br />
14 m 1 _ ; positive<br />
2<br />
15 m 3; positive<br />
16 a y 5<br />
b y 1<br />
c x 2<br />
d x 10<br />
e y 4.5<br />
8-1 The Slope of a Line 39
g e<br />
17 a _<br />
f d<br />
b Undefined<br />
c a _<br />
b<br />
d 1<br />
18 m 3 _<br />
2<br />
19 y 3<br />
20 y 14<br />
21 y 4<br />
22 x 0<br />
23 x 15<br />
24 x 8<br />
25 x 4<br />
26 a m 1<br />
b m 1<br />
27 a–c Not collinear<br />
28 m 4<br />
29 15 feet<br />
30 a m −− <br />
1<br />
AB _ , m<br />
3 −−−<br />
CD<br />
b Not collinear<br />
31 m −−<br />
AB<br />
<br />
2 _<br />
<br />
1<br />
_<br />
, m<br />
3 −−−<br />
AD<br />
<br />
4<br />
_<br />
11<br />
, m<br />
3 −− is undefined, m −−− 0. The<br />
BC AD<br />
triangle is a right triangle because −−<br />
BC is<br />
parallel to <strong>the</strong> x-axis and −−<br />
CA is parallel to <strong>the</strong><br />
y-axis, making <strong>the</strong>se sides perpendicular to<br />
each o<strong>the</strong>r.<br />
8-2 The Equation of a Line<br />
(page 147)<br />
1 a No<br />
b Yes<br />
c Yes<br />
2 a y x 6; m 1, b 6<br />
b y 3x 5; m 3, b 5<br />
c y 3x; m 3, b 0<br />
d y 2x 4; m 2, b 4<br />
e y 2 _ x 4; m <br />
3 2 _ , b 4<br />
3<br />
f y 3 _ x 3; m <br />
2 3 _ , b 3<br />
2<br />
g y 3x 5; m 3, b 5<br />
h y 1 _ x <br />
2 9 _ ; m <br />
2 1 _ , b <br />
2 9 _<br />
2<br />
3 a x-intercept 7, y-intercept 7<br />
b x-intercept 4, y-intercept 8<br />
c x-intercept 3, y-intercept 6<br />
40 Chapter 8: Slopes and Equations of Lines<br />
d x-intercept 2, y-intercept 4<br />
e x-intercept 0, y-intercept 0<br />
f x-intercept 6, y-intercept 4<br />
4 a y 2x 5<br />
b y 1 _ x 2<br />
2<br />
c y 3<br />
d y x 4<br />
5 a y x 3<br />
b y 2<br />
c y 1 _ x 5<br />
2<br />
d y 5 _ x 3<br />
2<br />
e y 3 _ x 2<br />
2<br />
f y 3 _ x 5<br />
5<br />
6 a y 7 _ x 5<br />
2<br />
b y 2x 2<br />
c y 5x 10<br />
d y 3 _ x 3<br />
2<br />
e y 4x 13<br />
f y 6 _ x 6<br />
5<br />
7 a y x 1<br />
b y 2 _ x 2<br />
3<br />
c y 5<br />
d y 2x 1<br />
e y 2x 11<br />
f y 1 _ x <br />
3 1 _<br />
3<br />
8-3 The Slopes of Parallel<br />
and Perpendicular Lines<br />
(page 150)<br />
1 (4) Line l has a negative slope.<br />
2 (1) 5<br />
3 a Slope parallel 9 _<br />
7<br />
Slope perpendicular 7 _<br />
9<br />
b Slope parallel 0<br />
No slope <strong>for</strong> perpendicular line<br />
c Slope parallel 7 _<br />
8<br />
Slope perpendicular 8 _<br />
7
d Slope parallel 2 _<br />
5<br />
Slope perpendicular 5 _<br />
e Slope parallel 9 _<br />
10<br />
Slope perpendicular 10 _<br />
9<br />
f Slope parallel 13<br />
Slope perpendicular 1 _<br />
13<br />
4 a Perpendicular; slopes are negative<br />
reciprocals<br />
b Perpendicular; slopes are negative<br />
reciprocals<br />
c Nei<strong>the</strong>r; slopes are nei<strong>the</strong>r equal nor<br />
negative reciprocals<br />
d Nei<strong>the</strong>r; slopes are nei<strong>the</strong>r equal nor<br />
negative reciprocals<br />
e Perpendicular; lines with a slope of 0 are<br />
perpendicular to lines with no slope<br />
f Parallel; slopes are <strong>the</strong> same<br />
5 a Yes<br />
b Yes<br />
c No<br />
6 a y 1 _ x <br />
4 17 _<br />
4<br />
b y 2x 7<br />
c y 1 _ x 4<br />
5<br />
7 a Right triangle<br />
b Not a right triangle<br />
8-4 The Midpoint of a Line<br />
Segment<br />
(pages 153–154)<br />
1 (2) (1.5, 1)<br />
2 (4) (2.5, 2)<br />
3 (2) (2, 0.1)<br />
4 (1) (8, 0)<br />
5 (4) (17, 20)<br />
6 (4) (13, 13)<br />
7 (5a, 5r)<br />
8 (3x, 5y)<br />
9 (10, 11)<br />
2<br />
10<br />
y<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
G<br />
1<br />
8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8<br />
1<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
8<br />
D<br />
L<br />
A<br />
A rectangle is <strong>for</strong>med.<br />
M GL (0, 1), M LA (2, 1), M DA (2, 3),<br />
M GD (0, 3)<br />
11 (0, 2)<br />
12 a −−<br />
BD is parallel to −−<br />
EF because <strong>the</strong>ir slopes<br />
are 3 _ .<br />
4<br />
b EF 1 _ BD<br />
2<br />
5 1 _ (10)<br />
2<br />
5 5<br />
13 a m 5 _<br />
2<br />
b m 2 _<br />
5<br />
5<br />
c M ( 4, _<br />
2 )<br />
d y 5 _ x <br />
6 35 _<br />
6<br />
14 y 4x 9<br />
15 y 1 _ x <br />
5 4 _<br />
5<br />
8-5 Coordinate Proof<br />
(pages 163–165)<br />
1 The triangle is an isosceles triangle because<br />
WI WN √ 40 .<br />
2 Slopes of −−−<br />
NA and −−−<br />
AQ are negative reciprocals<br />
proving <strong>the</strong>se two lines <strong>for</strong>m a right<br />
angle by being perpendicular; <strong>the</strong>re<strong>for</strong>e,<br />
NAQ is a right triangle.<br />
3 AM CM BM √ 26 ; <strong>the</strong> midpoint of <strong>the</strong><br />
hypotenuse is equidistant from all three<br />
vertices.<br />
4 The slopes are negative reciprocals so <strong>the</strong><br />
median is also <strong>the</strong> altitude.<br />
x<br />
8-5 Coordinate Proof 41
5 BIG is isosceles because it has two congruent<br />
sides, BI IG √ 50 ; and BIG is<br />
a right triangle because −−<br />
BI is perpendicular<br />
to −−<br />
IG .<br />
6 Two sides ( −−−<br />
QR and −−<br />
PS ) have <strong>the</strong> same slope<br />
of 2 _ and <strong>the</strong> o<strong>the</strong>r two sides (<br />
3 −−<br />
OP and −−<br />
RS )<br />
have <strong>the</strong> same slope of 3. PQRS is a<br />
parallelogram.<br />
7 SAND is a parallelogram because <strong>the</strong> diagonals<br />
bisect each o<strong>the</strong>r at <strong>the</strong> point (1.5, 0.5).<br />
8 LEAP is a parallelogram because all of <strong>the</strong><br />
sides measure √ 50 and are congruent.<br />
9 ABCD is a parallelogram because <strong>the</strong> diagonals<br />
bisect each o<strong>the</strong>r at <strong>the</strong> point (2.5, 0.5).<br />
10 a 6 √ 5 and 2 √ 13<br />
b The diagonals meet at <strong>the</strong>ir midpoint<br />
(1, 1).<br />
c BETH is a parallelogram because <strong>the</strong><br />
diagonals bisect each o<strong>the</strong>r.<br />
d BETH is not a rectangle because <strong>the</strong><br />
diagonals are not congruent.<br />
11 a NICK is a parallelogram; both pairs of<br />
opposite sides are parallel.<br />
b NICK is not a rhombus; <strong>the</strong> diagonals are<br />
not perpendicular.<br />
12 The figure KATE is a square. KATE is a rectangle<br />
because <strong>the</strong> diagonals bisect each o<strong>the</strong>r<br />
and are <strong>the</strong> same length. KATE is a square<br />
because two adjacent sides are congruent.<br />
13 ABCD is a rhombus. ABCD is a parallelogram<br />
because <strong>the</strong> diagonals bisect each o<strong>the</strong>r.<br />
ABCD is a rhombus because two adjacent<br />
sides are congruent.<br />
14 a Slopes of −−<br />
SU and −−<br />
UE are negative reciprocals<br />
proving <strong>the</strong>se two lines <strong>for</strong>m a right<br />
angle by being perpendicular; <strong>the</strong>re<strong>for</strong>e,<br />
SUE is a right triangle.<br />
b SU 5 and UE 10<br />
15 a NORA is a parallelogram because <strong>the</strong><br />
diagonals bisect each o<strong>the</strong>r. NORA is a<br />
rhombus because two adjacent sides are<br />
congruent.<br />
b NORA is not a square because it does not<br />
contain a right angle.<br />
42 Chapter 8: Slopes and Equations of Lines<br />
16 a JACK is a trapezoid because it has only<br />
one pair of opposite sides parallel;<br />
slope of −−<br />
JA slope of −−<br />
CR 1.<br />
b JACK is not an isosceles trapezoid because<br />
<strong>the</strong> legs are not congruent.<br />
17 MARY is a trapezoid because it has only<br />
one pair of opposite sides parallel; slope of<br />
−−−<br />
MA slope of −−<br />
RY 0. MARY is an isosceles<br />
trapezoid because <strong>the</strong> legs are congruent;<br />
AR YM 5.<br />
18 C(a, 0)<br />
19 P(b a, c)<br />
20 E(a, 0), F(a, 2a), G(a, 2a)<br />
21 C(a, b r)<br />
22 M(a c, b)<br />
23 T(a, 2a)<br />
24 R(a b, c)<br />
25 Yes<br />
26 Slope of −−<br />
AC is 1 and <strong>the</strong> slope of −−<br />
BD is 1.<br />
Since <strong>the</strong> slopes are negative reciprocals,<br />
−−<br />
AC −−<br />
BD .<br />
27 a Midpoint of −−<br />
BC is (a, b).<br />
28 a<br />
b MA MB MC √ a 2 b 2<br />
T(0, 0)<br />
y<br />
E(2x, 2y)<br />
Q(4x, 0)<br />
x<br />
b A(x, y), B(3x, y), C(2x, 0)<br />
c The length of median −−<br />
TB is √ 9 x 2 y 2 .<br />
The length of median −−<br />
EC is 2y. The length<br />
of median −−−<br />
QA is √ 9 x 2 y 2 .<br />
d TEQ is an isosceles triangle.<br />
29 TA TR √ <br />
( c _<br />
2 ) 2<br />
d 2<br />
30 −−<br />
JA and −−<br />
NE are parallel to <strong>the</strong> x-axis and<br />
each slope is 0; thus <strong>the</strong>y are parallel to each<br />
o<strong>the</strong>r. Slope of −−−<br />
AN slope of −−<br />
EJ c _ ; thus<br />
b<br />
<strong>the</strong>y are parallel. There<strong>for</strong>e, JANE is a<br />
parallelogram.<br />
31 Midpoint of −−<br />
AC midpoint of<br />
−−<br />
BD <br />
a r ( _ ,<br />
s<br />
2 _<br />
2 )<br />
32 TA EM √ <br />
(a b) 2 c 2
8-6 Concurrence of <strong>the</strong><br />
Altitudes of a Triangle<br />
(<strong>the</strong> Orthocenter)<br />
(page 168)<br />
1 (4) obtuse<br />
2 (4) at one of <strong>the</strong> vertices of <strong>the</strong> triangle<br />
Exercises 3–9: Check students’ graphs.<br />
3 (2.2, 1.2)<br />
5<br />
4 ( _ , 4) 3<br />
5 (1, 0)<br />
6 (6, 6.5)<br />
7 ( 2,<br />
22 _<br />
3 )<br />
8 (1, 1)<br />
9 (1, 0.5)<br />
10 a Slope of −−<br />
AB is 4 _ . Slope of<br />
3 −−<br />
BC is 3 _ . Since<br />
4<br />
<strong>the</strong> slopes are negative reciprocals, <strong>the</strong><br />
segments are perpendicular and B is <strong>the</strong><br />
right angle.<br />
b −−<br />
AB and −−<br />
BC can each be altitudes. B is <strong>the</strong><br />
endpoint of <strong>the</strong> altitude drawn from B<br />
to AC.<br />
Chapter Review (pages 168–171)<br />
1 (2) y 3 _ x 4<br />
4<br />
2 (2) (0, 3)<br />
3 (4) (1, 9)<br />
4 (2) y 5x 1<br />
5 (1) y 3x 3<br />
6 No. Sub-in: 3(5) 4(2) 0<br />
7 x-intercept is (2, 0) and y-intercept is (0, 3).<br />
Distance is √ 13 .<br />
8 x-intercept is (15, 0) and y-intercept is (0, 3).<br />
Distance is 15.3.<br />
9 a 10<br />
10 B(10, 10)<br />
11 Slope of ___<br />
OB is 1 _ . Slope of<br />
2 ___<br />
AC is 2. Their<br />
product is ( 1 _<br />
2 ) (2) 1.<br />
12 a (0, 3) b (4, 0) c (8, 3)<br />
13 a m 3 b x-intercept is (4, 0)<br />
c y-intercept is (0, 12)<br />
d (3, 3) is not on <strong>the</strong> line.<br />
14 a nei<strong>the</strong>r b vertical c vertical<br />
d nei<strong>the</strong>r e horizontal f vertical<br />
15 a 1 _ b 2 c<br />
2 3 _<br />
4<br />
d 2 _ e <br />
3<br />
2 _ f <br />
3<br />
a _<br />
d<br />
16 a no slope b 0 c 1<br />
d 7 _ e<br />
d b<br />
2 _<br />
c a<br />
f<br />
k _<br />
h<br />
17 a 7 b 2 c 14 d 2<br />
e 6 f 5 g 4<br />
18 a not collinear<br />
b collinear<br />
c collinear<br />
d collinear<br />
e collinear<br />
f collinear<br />
19 a y 3<br />
b y 3 _ x <br />
2 9 _<br />
2<br />
c y 1 _ x <br />
3 5 _<br />
3<br />
d y 2 _ x<br />
3<br />
e y 3 _ x <br />
2 19 _<br />
2<br />
20 a AB AC 2 √ 10<br />
b Midpoint of ___<br />
BC is (2, 1). Slope of <strong>the</strong><br />
median from A to ___<br />
BC slope of altitude<br />
from A to ___<br />
BC 1.<br />
21 a (3, 2) b (2, 3) c (6.5, 8.5)<br />
22 Slope of ___<br />
DR slope of ____<br />
AW 1 _<br />
. Slope of<br />
3 ___<br />
RA is not equal to <strong>the</strong> slope of ____<br />
DW .<br />
___<br />
RA ____<br />
DW 5<br />
23 a NO 5 √ 2 and OP 2 √ 5<br />
b Slope of ___<br />
EO is 1 _ . Slope of<br />
3 ___<br />
PN is 3.<br />
Slopes of perpendicular lines are negative<br />
reciprocals of each o<strong>the</strong>r.<br />
24 A(0, 3), B(4, 2), C(5, 4), and D(1, 3). ABCD<br />
is a parallelogram because both pairs of opposite<br />
sides are parallel. Slope of ___<br />
AB slope<br />
of ___<br />
CD 1 _ . Slope of<br />
4 ___<br />
BC slope of ___<br />
DA 6.<br />
25 QRST is a parallelogram because both pairs<br />
of opposite sides are parallel. Slope of<br />
___<br />
QR slope of ___<br />
ST b _<br />
a<br />
. Slope of ___<br />
RS slope<br />
of ___<br />
TQ 0.<br />
26 a–b Check students’ graphs.<br />
c (2, 10)<br />
Chapter Review 43
___ 27 a M KA (5, 1), M ___<br />
AT (4, 3), M ___<br />
TK (2, 1)<br />
___ b m KA 1, m ___<br />
AT 0, m ___<br />
TK 2<br />
c Slope of line perpendicular to ___<br />
KA is 1.<br />
There is no slope <strong>for</strong> <strong>the</strong> line perpendicular<br />
to ___<br />
AT . Slope of <strong>the</strong> line perpendicular<br />
to ___<br />
TK is 1 _ .<br />
2<br />
d Perpendicular bisector of ___<br />
KA : y x 6<br />
Perpendicular bisector of ___<br />
AT : x 4<br />
Perpendicular bisector of ___<br />
TK : y 1 _ x<br />
2<br />
e (4, 2)<br />
Parallel Lines<br />
9-2 Proving Lines Parallel<br />
(pages 177–179)<br />
1 none<br />
2 a b, c d<br />
3 none<br />
4 b c<br />
5 a b<br />
6 a c<br />
7 b c<br />
8 l m<br />
9 a and b<br />
10 c and d<br />
11 a and b<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
44 Chapter 9: Parallel Lines<br />
28 a y 2x 5<br />
b (6, 8)<br />
c y 1 _ x 5<br />
2<br />
CHAPTER<br />
9<br />
12 1. 1 3<br />
2. 2 4<br />
3. 1 2 3 4 (Addition postulate<br />
of equality)<br />
4. ABG DEG<br />
5. −−<br />
ED −−<br />
BA (If two lines are cut by a transversal<br />
<strong>for</strong>ming a pair of congruent<br />
alternate interior angles,<br />
<strong>the</strong> two lines are parallel.)<br />
13 1. ___<br />
AF ___<br />
CD<br />
2. −−<br />
FC ___<br />
FC<br />
3. −−<br />
AF −−<br />
FC ___<br />
CD ___<br />
FC<br />
4. ___<br />
AC ___<br />
DF<br />
5. −−<br />
BC ___<br />
EF<br />
6. ___<br />
BC ___<br />
AD<br />
7. BCF is a right angle. (Definition of<br />
right angle)<br />
8. ___<br />
EF ___<br />
AD<br />
9. EFD is a right angle.<br />
10. BCF EFD (Right angles are<br />
congruent)<br />
11. ABC DEF (SAS SAS)
12. BAC EDF (CPCTC)<br />
13. ___<br />
ED ___<br />
AD (Alternate interior angles are<br />
congruent)<br />
14 1. ___<br />
BE bisects ABC.<br />
2. 1 EBC<br />
3. m1 mEBC<br />
4. ___<br />
CE bisects DCB.<br />
5. 2 ECB<br />
6. m2 mECB<br />
7. m1 m2 90<br />
8. mEBC mECB 90 (Substitution<br />
postulate)<br />
9. m1 m2 mEBC mECB<br />
10. m1 mEBC 90<br />
11. m2 mECB 90<br />
12. ABC and DCB are right angles.<br />
13. BA ___<br />
BC<br />
14. CD ___<br />
BC<br />
15. BA CD (Two lines perpendicular to <strong>the</strong><br />
same line are parallel.)<br />
15 1. 1 3<br />
2. m1 m3<br />
3. 2 4<br />
4. m2 m4<br />
5. m1 m2 m3 m4 360<br />
6. m1 m1 m2 m2 360<br />
or 2(m1) 2(m2) 360<br />
7. m1 m2 180 (Division<br />
postulate)<br />
8. −−−<br />
AD −−<br />
BC (Interior angles on <strong>the</strong> same<br />
side of <strong>the</strong> transversal are<br />
supplementary.)<br />
9. m1 m1 + m4 m4 360<br />
or 2(m1) 2(m4) 360<br />
10. m1 m4 180<br />
11. ___<br />
AB ___<br />
CD<br />
16 1. D is <strong>the</strong> midpoint of ___<br />
CF and of ___<br />
BE .<br />
2. ___<br />
CD ___<br />
DF<br />
3. ___<br />
BD ___<br />
DE<br />
4. ___<br />
FC ___<br />
FC<br />
5. CBD FDE (Vertical<br />
angles are<br />
congruent.)<br />
6. CBD FDE (SAS SAS)<br />
7. EFD DCB (CPCTC)<br />
8. ___<br />
AC ___<br />
FE (Alternate interior angles are<br />
congruent.)<br />
17 1. 1 2<br />
2. AFC DCF (Supplementary angles<br />
of congruent angles are<br />
congruent.)<br />
3. ___<br />
EF ___<br />
CB<br />
4. ___<br />
FC ___<br />
FC<br />
5. ___<br />
EF ___<br />
FC = ___<br />
BC ___<br />
FC<br />
6. ___<br />
EC ___<br />
BF<br />
7. ___<br />
AF ___<br />
CD<br />
8. AFB DCE (SAS SAS)<br />
9. ABF DEC (CPCTC)<br />
10. ___<br />
AB ___<br />
ED (Alternate interior angles<br />
are congruent.)<br />
18 1. ___<br />
AE ___<br />
FC<br />
2. ___<br />
EF ___<br />
EF<br />
3. ___<br />
AE ___<br />
EF ___<br />
FC ___<br />
EF<br />
4. ___<br />
AF ___<br />
CE<br />
5. ___<br />
DE ___<br />
AC<br />
6. DEA is a right angle.<br />
7. ___<br />
BF ___<br />
AC<br />
8. BFC is a right angle.<br />
9. DEA BFC (Right angles are<br />
congruent.)<br />
10. ___<br />
DE ___<br />
BF<br />
11. AFB CED (SAS SAS)<br />
12. ______<br />
AEFC<br />
13. BAF DCE (CPCTC)<br />
AB ___<br />
DC<br />
14. ___<br />
9-3 Properties of Parallel<br />
Lines<br />
(pages 182–185)<br />
1 (1) same-side exterior angles<br />
2 (2) 2, 3, 6, 7, 10, 11, 14, 15<br />
3 m1 m4 m6 m7 60;<br />
m2 m3 m5 120<br />
4 m1 m4 m5 m8 135;<br />
m2 m3 m6 m7 45<br />
5 ma md mg 65;<br />
mb mc me mf 125<br />
6 w y 70; x z 110<br />
7 a 8x 6x 3 0 180<br />
x 15<br />
b 2x 1 0 5x 47<br />
x 19<br />
m3 2(19) 10 48<br />
9-3 Properties of Parallel Lines 45
8 106<br />
9 mA 75, mC 67<br />
10 57<br />
11 60<br />
12 1. Perpendicular lines <strong>for</strong>m right angles.<br />
2. Right angles measure 90.<br />
3. When two parallel lines are cut by a<br />
transversal, corresponding angles are<br />
congruent.<br />
4. Congruent angles have equal measure.<br />
5. Transitive property of congruence (3, 5)<br />
6. If an angle measures 90, it is a right<br />
angle.<br />
7. If two lines intersect to <strong>for</strong>m a right angle,<br />
<strong>the</strong>y are perpendicular.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
13 1. ___<br />
AB<br />
2. 2 is supplement of 1.<br />
3. 2 is supplement of 3.<br />
4. 1 3 (If two angles are supplements<br />
of <strong>the</strong> same angle, <strong>the</strong>n <strong>the</strong>y<br />
are congruent.)<br />
5. ___<br />
BC ___<br />
AD (When two lines are cut by<br />
a transversal creating corresponding<br />
angles, <strong>the</strong> lines are<br />
congruent.)<br />
14 1. ABC<br />
2. ___<br />
AB ___<br />
BC<br />
3. BAC BCA (Isosceles triangle<br />
<strong>the</strong>orem)<br />
4. _____<br />
FHD ___<br />
BC<br />
5. FDA BCA (Corresponding<br />
exterior angles are<br />
congruent.)<br />
6. _____<br />
EHG ___<br />
AB<br />
7. BAC GEC (Corresponding<br />
exterior angles are<br />
congruent.)<br />
8. GEC FDA<br />
9. ___<br />
HE ____<br />
HD<br />
10. EHD is isosceles. (Definition of<br />
isosceles triangle)<br />
46 Chapter 9: Parallel Lines<br />
15 1. ___<br />
DE ___<br />
AC<br />
2. 1 2 (Corresponding exterior<br />
angles are congruent.)<br />
3. 3 4<br />
4. 2 3<br />
5. 1 4 (Transitive property of<br />
congruence)<br />
16 1. ___<br />
AC intersects ___<br />
BD at E.<br />
2. ___<br />
AE ___<br />
ED<br />
3. A D<br />
4. ___<br />
AD ___<br />
BC<br />
5. A C (Alternate interior angles are<br />
congruent.)<br />
6. B D (Alternate interior angles are<br />
congruent.)<br />
7. B C (Transitive property of<br />
congruence)<br />
8. ___<br />
AE ___<br />
CE (Isosceles triangle<br />
<strong>the</strong>orem)<br />
9. ___<br />
BE ___<br />
DE<br />
10. ___<br />
AC ___<br />
BD (Addition postulate)<br />
17 1. n m<br />
2. ABE and BAD are supplementary.<br />
(Two interior angles on <strong>the</strong><br />
same side of <strong>the</strong> transversal<br />
are supplementary.)<br />
3. mABE mBAD 180<br />
4. 1 _ mABE <br />
2 1 _ mBAD 90<br />
2<br />
(Division postulate)<br />
5. ___<br />
BC bisects ABE.<br />
6. mABC 1 _ mABE<br />
2<br />
(Definition of angle bisector)<br />
7. ___<br />
AC bisects BAD.<br />
8. mACB 1 _ mBAD<br />
2<br />
( Definition of angle bisector)<br />
9. mABC mACB 90<br />
10. mBCA mABC mACB 180<br />
11. mBCA 90<br />
12. BCA is a right angle.<br />
13. ___<br />
BC ___<br />
AC (Perpendicular lines <strong>for</strong>m<br />
right angles.)<br />
18 1. r m and a b<br />
2. 4 and 3 are supplementary.<br />
3. 2 and 3 are supplementary.<br />
4. (a) 4 2 (Supplements of <strong>the</strong> same<br />
angle are congruent.)<br />
5. 4 and 5 are supplementary.
6. 2 and 1 are supplementary.<br />
7. (b) 5 1 (Supplements of <strong>the</strong> same<br />
angle are congruent.)<br />
19 a (x 12) (3x) 180<br />
x 42<br />
m1 m3 m5 54<br />
m2 m4 m6 126<br />
b (2x) (2x 20) 180<br />
x 50<br />
m1 m3 m5 100<br />
m2 m4 m6 80<br />
c (7x 65) (5x 5)<br />
x 30<br />
m2 m5 145<br />
m1 m3 m4 m6 35<br />
20 a 5<br />
b 7<br />
c 6 and 8<br />
d m5 60<br />
e m4 90<br />
21 Since <strong>the</strong> corresponding angles are congruent,<br />
<strong>the</strong> halves of each are congruent. They<br />
<strong>for</strong>m new congruent corresponding angles<br />
cut by <strong>the</strong> same transversal. The bisectors are<br />
<strong>the</strong>re<strong>for</strong>e parallel.<br />
22 Draw a diagonal line, <strong>for</strong>ming two congruent<br />
triangles by SAS. The o<strong>the</strong>r two sides of<br />
<strong>the</strong> quadrilateral and remaining angles are<br />
congruent by CPCTC. There<strong>for</strong>e, since <strong>the</strong><br />
diagonal is a transversal <strong>for</strong> <strong>the</strong>se sides, <strong>the</strong><br />
sides are parallel.<br />
9-4 Parallel Lines in <strong>the</strong><br />
Coordinate Plane<br />
(pages 187–188)<br />
1 a 4<br />
b 7 _<br />
3<br />
c 2 _<br />
5<br />
d x _<br />
a<br />
2 a 1<br />
b 3 _<br />
5<br />
c 2 _<br />
5<br />
3 a 1 _<br />
4<br />
b 7<br />
c 1<br />
d 3 _<br />
2<br />
4 a perpendicular<br />
b parallel<br />
c perpendicular<br />
d parallel<br />
e perpendicular<br />
f nei<strong>the</strong>r<br />
5 a 1<br />
b 5 _<br />
2<br />
c 3 _<br />
10<br />
d 7 _<br />
9<br />
e 4 _<br />
3<br />
f 3 _<br />
2<br />
6 a 7<br />
b 1<br />
c 11<br />
d no slope<br />
e 0<br />
f 1 _<br />
2<br />
___<br />
7 a Slope AB 7, slope ___ <br />
1<br />
CD _ ,<br />
7 ___<br />
AB ___<br />
CD<br />
___ b Slope AB 2, slope ___ <br />
1<br />
CD _ ,<br />
2 ___<br />
AB ___<br />
CD<br />
8 y 5x 3<br />
9 y = 1 _ x 3<br />
5<br />
10 y 3 _ x <br />
2 7 _<br />
2<br />
11 y 1 _ x 3<br />
3<br />
12 y 4<br />
13 y 2x 6<br />
14 y 2 _ x 9<br />
3<br />
15 k = 1<br />
16 (7, 5)<br />
___ 17 Slope <br />
4<br />
AB _<br />
3<br />
___ , slope CD <br />
3 _ . The slopes are<br />
4<br />
negative reciprocals, <strong>the</strong>re<strong>for</strong>e −−<br />
AB and −−−<br />
CD<br />
are perpendicular and ABC is a right<br />
triangle.<br />
18 Opposite sides have <strong>the</strong> same slope.<br />
___ m PQ m ___ <br />
1<br />
RS _<br />
___ and m PS 3 m ___ 5 QR<br />
19 H(5, 6)<br />
20 a D(4, 3)<br />
9-4 Parallel Lines in <strong>the</strong> Coordinate Plane 47
9-5 The Sum of <strong>the</strong><br />
Measures of <strong>the</strong> Angles<br />
of a Triangle<br />
(pages 193–195)<br />
1 (4) scalene<br />
2 (3) obtuse<br />
3 (3) 120<br />
4 (2) right<br />
5 (3) 112<br />
6 a base angles are 30, vertex angle is 120<br />
b base angles are 60, vertex angle is 60<br />
c base angles are 52, vertex angle is 76<br />
d base angles are 36, vertex angle is 108<br />
e base angles are 20, vertex angle is 140<br />
7 a base angles are 50, exterior angle is 130<br />
b base angles are 40, exterior angle is 140<br />
c base angles are 54, exterior angle is 126<br />
d base angles are 60, exterior angle is 120<br />
e base angles are 80, exterior angle is 100<br />
8 base angles are 74, vertex angle is 32<br />
9 mP 27, mQ 45, mR 108<br />
10 base angles are 28, vertex angle is 124.<br />
11 exterior angle is 30, base angles are 15,<br />
vertex angle is 150<br />
12 18, 54, 108<br />
13 99, 45, 36<br />
14 mc 35<br />
15 mB 78<br />
16 mx 150<br />
17 mx 30<br />
18 m1 150<br />
19 mD 20<br />
20 md 125<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
21 1. A C<br />
2. ___<br />
BD bisects ABC.<br />
3. ABD CBD<br />
4. ___<br />
BD ___<br />
BD<br />
5. ABD CBD (AAS AAS)<br />
48 Chapter 9: Parallel Lines<br />
6. ADB CDB (CPCTC)<br />
7. ADB is supplementary to CDB.<br />
8. ADB and CDB are right angles.<br />
9. −−<br />
BD −−<br />
AC (Segments <strong>for</strong>ming right<br />
angles are perpendicular.)<br />
22 1. A C<br />
2. ___<br />
BD ___<br />
AC<br />
3. BDA and BDC are right angles.<br />
4. BDA BDC<br />
5. ___<br />
BD ___<br />
BD<br />
6. BDA DBC (AAS AAS)<br />
7. ABD CBD (CPCTC)<br />
8. ___<br />
BD bisects ABC. (Definition of angle<br />
bisector)<br />
23 Both ABC and DEC share C. Since <strong>the</strong><br />
remaining angles are <strong>the</strong> same as well, 1<br />
and B are congruent corresponding angles.<br />
There<strong>for</strong>e, ___<br />
BA ___<br />
ED .<br />
24 Compare ABC and DEC. Both are right<br />
triangles with one pair of congruent acute<br />
angles. There<strong>for</strong>e, <strong>the</strong> o<strong>the</strong>r pair of acute<br />
angles are also congruent, B CED. But<br />
CED 1, vertical pairs. Then B 1<br />
by <strong>the</strong> transitive postulate of congruence.<br />
9-6 Proving Triangles<br />
Congruent by Angle,<br />
Angle, Side<br />
(pages 198–200)<br />
1 b and f<br />
2 a Not sufficient. If <strong>the</strong> third side is congruent,<br />
SSS. If included angles are congruent,<br />
SAS.<br />
b Sufficient, SSS<br />
c Sufficient, hypotenuse-angle<br />
d Not sufficient. If ei<strong>the</strong>r pair of corresponding<br />
angles are congruent, AAS.<br />
e Sufficient, AAS<br />
f Not sufficient. If any pair of corresponding<br />
sides are congruent, AAS.<br />
g Sufficient, hypotenuse-leg<br />
h Sufficient, SAS
i Sufficient, SSS<br />
j Sufficient, AAS<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
3 1. ___<br />
AB ___<br />
EF<br />
2. ABC FED<br />
3. ___<br />
BC ___<br />
DE<br />
4. 1 2<br />
5. I II (ASA ASA)<br />
4 1. 1 4<br />
2. B D<br />
3. ___<br />
AC ___<br />
AC<br />
4. I II (AAS AAS)<br />
5 1. ___<br />
AB ___<br />
DE<br />
2. DEC BAC (Alternate interior<br />
angles are congruent.)<br />
3. ABC EDC (Alternate interior<br />
angles are congruent.)<br />
4. C is <strong>the</strong> midpoint of ___<br />
BD .<br />
5. ___<br />
BC ___<br />
CD (Definition of<br />
midpoint)<br />
6. ABC EDC (AAS AAS)<br />
6 1. B D<br />
2. BEC DEA<br />
3. ___<br />
BC ___<br />
AD<br />
4. AED CEB (AAS AAS)<br />
5. ___<br />
AE ___<br />
CE (CPCTC)<br />
7 1. A E<br />
2. ___<br />
BC ___<br />
DC<br />
3. ___<br />
AC ___<br />
CE<br />
4. ___<br />
AC ___<br />
BC (Subtraction<br />
___<br />
CE ___<br />
DC postulate)<br />
or ___<br />
AB ___<br />
DE<br />
5. ___<br />
BG ___<br />
AE<br />
6. BGA is a right (Definition of<br />
angle. right angle)<br />
7. ___<br />
DF ___<br />
AE<br />
8. ____<br />
DFE is a right angle.<br />
9. BGA DFE (Right angles are<br />
congruent.)<br />
10. BGA DFE (AAS AAS)<br />
BG ___<br />
DF (CPCTC)<br />
11. ___<br />
8 1. ___<br />
AB ___<br />
EF<br />
2. ABC FED<br />
3. A F<br />
4. ___<br />
AC ___<br />
DF<br />
5. I II (AAS AAS)<br />
9 1. ___<br />
AB ___<br />
CD<br />
2. ___<br />
CD ___<br />
AB<br />
3. CDA is a right angle.<br />
4. ___<br />
AE ___<br />
CB<br />
5. AEC is a right angle.<br />
6. BAC BCA (Isosceles triangle<br />
<strong>the</strong>orem)<br />
7. ___<br />
AC ___<br />
AC<br />
8. CDA AEC (AAS AAS)<br />
9. ___<br />
CD ___<br />
AE (CPCTC)<br />
10 1. E is <strong>the</strong> midpoint of ___<br />
AC .<br />
2. ___<br />
AE ___<br />
CE<br />
3. ___<br />
AF ___<br />
BD<br />
4. AFE is a right angle.<br />
5. ___<br />
CD ___<br />
BD<br />
6. CDE is a right angle.<br />
7. AFE CDE<br />
8. CED AEF (Vertical angles are<br />
congruent.)<br />
9. AFE CDE (AAS AAS)<br />
10. ___<br />
AF ___<br />
CD (CPCTC)<br />
11 1. ___<br />
QR ___<br />
SR<br />
2. RQS RSQ (Isosceles triangle<br />
<strong>the</strong>orem)<br />
3. 1 2<br />
4. ___<br />
QS ___<br />
QS<br />
5. QTS SMQ (AAS AAS)<br />
6. ____<br />
QM ___<br />
ST (CPCTC)<br />
12 1. ___<br />
BA ___<br />
CD<br />
2. ___<br />
BA ___<br />
AD<br />
3. BAQ is a right angle.<br />
4. ___<br />
CD ___<br />
AD (Two lines perpendicular<br />
to <strong>the</strong> same line<br />
are parallel.)<br />
5. CDP is a right angle.<br />
6. BAQ CDP<br />
7. B C<br />
8. ___<br />
AP ____<br />
QD<br />
9. ___<br />
PQ ___<br />
PQ<br />
10. ___<br />
AD ___<br />
PD (Addition postulate)<br />
11. BAQ CDP (AAS AAS)<br />
BA ___<br />
CD (CPCTC)<br />
12. ___<br />
9-6 Proving Triangles Congruent by Angle, Angle, Side 49
9-7 The Converse of <strong>the</strong><br />
Isosceles Triangle Theorem<br />
(pages 202–203)<br />
1 2(3x 4) 2x 4 180<br />
x 22<br />
mA mC 70<br />
mB 40<br />
2 2(3x 3) 4x 16 180<br />
x 19<br />
mA mB mC 60<br />
3 2x 180 82 82<br />
x 8<br />
4 (2x 14) (3x 2) (5x 8) 180<br />
x 16<br />
2x 14 2(16) 14 46<br />
3x 2 3(16) 2 46<br />
5x 8 5(16) 8 88<br />
Since two angles are equal in measure,<br />
ADC is isosceles.<br />
5 mA mB 49<br />
mx 180 49 49<br />
my 49<br />
mz 180 82 98<br />
6 mQ 58<br />
mx mz 64<br />
my 116<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
7 1. A C<br />
2. ___<br />
BC ___<br />
AB<br />
3. ___<br />
AD ___<br />
EC<br />
4. ABD CBE (SAS SAS)<br />
8 1. ___<br />
BD ___<br />
AC<br />
2. 1 ACB<br />
3. 2 BAC<br />
4. 1 2<br />
5. ACB BAC (Substitution<br />
postulate)<br />
6. ABC is isosceles. (Definition of<br />
isosceles triangle)<br />
50 Chapter 9: Parallel Lines<br />
9 1. 1 2<br />
2. 3 4 (Supplements of<br />
congruent angles are<br />
congruent.)<br />
3. ___<br />
AB ___<br />
BC<br />
4. ABC is isosceles. (Definition of<br />
isosceles triangle)<br />
10 1. ___<br />
BD ___<br />
AE<br />
2. ___<br />
AC ___<br />
CE<br />
3. A E<br />
4. A B (Two parallel lines are cut<br />
by a transversal <strong>the</strong>n <strong>the</strong><br />
corresponding angles are<br />
congruent.)<br />
5. E D (Two parallel lines are cut<br />
by a transversal <strong>the</strong>n <strong>the</strong><br />
corresponding angles are<br />
congruent.)<br />
6. B D (Transitive postulate of<br />
congruence)<br />
7. ___<br />
BC ___<br />
DC (Congruent angles imply<br />
congruent sides.)<br />
11 1. ___<br />
BC ___<br />
BD<br />
2. BCD BDC<br />
3. BDA BDE (Linear pairs of<br />
congruent angles)<br />
4. BAC BED (ASA ASA)<br />
5. ___<br />
AB ___<br />
BE (CPCTC)<br />
12 1. ___<br />
AB ___<br />
EB<br />
2. BAE BEA<br />
3. ___<br />
BD ___<br />
AE<br />
4. BAE 1<br />
5. BAE 2<br />
6. 1 2 (Substitution<br />
postulate)<br />
13 1. ___<br />
PQ ___<br />
SR<br />
2. ___<br />
PQ ___<br />
SR<br />
3. QPR SRP (Alternate interior<br />
angles are<br />
congruent.)<br />
4. ___<br />
PR bisects QPS.<br />
5. QRP RPS (Definition of<br />
bisector)<br />
6. SRP RPS (Transitive postulate<br />
of congruence)<br />
7. ___<br />
PS ___<br />
SR (Converse of <strong>the</strong><br />
isosceles triangle<br />
<strong>the</strong>orem)
14 1. ___<br />
DF ___<br />
FE<br />
2. 2 3<br />
3. 1 4<br />
4. ADF CEF (Linear pairs of congruent<br />
angles)<br />
5. ADF CEF (ASA ASA)<br />
6. A C (CPCTC)<br />
7. ABC is an isosceles triangle. (Definition<br />
of an isosceles triangle)<br />
15 1. ______<br />
AEDC<br />
2. 1 2<br />
3. ___<br />
BE ___<br />
BD<br />
4. AEB CDB<br />
5. ___<br />
AE ___<br />
DC<br />
6. ABE CDB (ASA ASA)<br />
7. A C (CPCTC)<br />
8. ABC is an isosceles triangle. (Definition<br />
of an isosceles triangle)<br />
9-8 Proving Right Triangles<br />
Congruent by Hypotenuse-<br />
Leg; Concurrence of Angle<br />
Bisectors of a Triangle<br />
(pages 207–208)<br />
1 (a)<br />
2 (d)<br />
3 (c)<br />
4 (b)<br />
5 (a)<br />
6 m1 24, m2 52, m3 104,<br />
m4 52, m5 14, m6 114,<br />
m7 14, m8 24, m9 142<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
7 1. ___<br />
SQ ___<br />
PR<br />
2. ___<br />
SA ___<br />
PB<br />
3. SQP and SAP are right angles.<br />
4. SQP and SAP are right triangles.<br />
SP ___<br />
SP<br />
5. ___<br />
6. ___<br />
SQ ___<br />
SA<br />
7. SQP SAP (HL HL)<br />
8. SPQ APS (CPCTC)<br />
9. ___<br />
PT bisects RPB. (Definition of<br />
angle bisector)<br />
8 1. ___<br />
AE ___<br />
BC<br />
2. ___<br />
CD ___<br />
AB<br />
3. CDA and CEA are right angles.<br />
4. CDA and CEA are right triangles.<br />
5. ___<br />
AC ___<br />
AC<br />
6. ___<br />
AE ___<br />
CD<br />
7. ACE CAD (HL HL)<br />
9 1. ___<br />
AE ___<br />
BC<br />
2. ___<br />
CD ___<br />
AB<br />
3. AEB and CDB are right angles.<br />
4. AEB and CDB are right triangles.<br />
5. ___<br />
DB ___<br />
EB<br />
6. DBE DBE<br />
7. AEB CDB (Leg–acute angle)<br />
8. ___<br />
AE ___<br />
CD (CPCTC)<br />
10 1. ___<br />
BD ___<br />
AC<br />
2. ___<br />
QS ___<br />
PR<br />
3. BDA and QSP are right angles.<br />
4. BDA and QSP are right triangles.<br />
5. ABC PQR<br />
6. ___<br />
AB ___<br />
PQ (CPCTC)<br />
7. A P (CPCTC)<br />
8. BDA QSP (Hypotenuse–<br />
acute angle)<br />
9. ABD PQS (CPCTC)<br />
10. mABD mPQS<br />
11. ___<br />
BD bisects ABC.<br />
12. 1 _ mABC mABD mPQS<br />
2<br />
13. 1 _ mABC <br />
2 1 _ mPQR (Division<br />
2<br />
postulate)<br />
14. mPQS 1 _ mPQR (Transitive<br />
2<br />
postulate)<br />
15. ___<br />
QS bisects PQR. (Definition of<br />
angle bisector)<br />
11 1. ___<br />
AB ___<br />
CF , ___<br />
DE ___<br />
CF<br />
2. ABF and CED are right angles.<br />
3. ABF and CED are right triangles.<br />
4. ___<br />
CD ___<br />
AF<br />
5. ___<br />
BE ___<br />
BE<br />
6. ___<br />
CE ___<br />
FB<br />
7. ABF CED (HL HL)<br />
AB ___<br />
DE<br />
8. ___<br />
9-8 Proving Right Triangles Congruent by Hypotenuse-Leg; Concurrence of Angle Bisectors of a Triangle 51
12 1. ___<br />
CE ___<br />
BA , ___<br />
BD ___<br />
AC<br />
2. CEA and BDA are right angles.<br />
3. CEA and BDA are right triangles.<br />
4. ___<br />
AB ___<br />
AC<br />
5. ABC ABC<br />
6. CEA BDA (Hypotenuse–acute<br />
angle)<br />
7. ___<br />
CE ___<br />
BD (CPCTC)<br />
13 1. ___<br />
AC ___<br />
BF , ___<br />
ED ___<br />
BF<br />
2. ACB and FDE are right angles.<br />
3. ACB and FDE are right triangles.<br />
4. ___<br />
AC ___<br />
ED<br />
5. ___<br />
BA ___<br />
EF<br />
6. ACB FDE (HL HL)<br />
7. A E (CPCTC)<br />
9-9 Interior and Exterior Angles of Polygons<br />
(pages 210–212)<br />
1 Polygon Number<br />
of Sides<br />
Number<br />
of Triangles<br />
Sum of<br />
Interior<br />
Angles<br />
180(n 2)<br />
Triangle 3 1 180(1) 180 180<br />
_<br />
3<br />
Quadrilateral 4 2 180(2) 360 360<br />
_<br />
4<br />
Pentagon 5 3 180(3) 540 540<br />
_<br />
5<br />
Hexagon 6 4 180(4) 720 720<br />
_<br />
6<br />
Heptagon 7 5 180(5) 900 900<br />
_<br />
7<br />
Octagon 8 6 180(6) 1,080 1,080<br />
_<br />
8<br />
Nonagon 9 7 180(7) 1,260 1,260<br />
_<br />
9<br />
Decagon 10 8 180(8) 1,440 1,440<br />
_<br />
10<br />
24-gon 24 22 180(22) 3,960 3,960<br />
_<br />
24<br />
n-gon n n 2 180(n 2)<br />
52 Chapter 9: Parallel Lines<br />
Measure of<br />
Each Exterior<br />
180(n 2)<br />
Angle _<br />
n<br />
Measure of<br />
Each Interior<br />
Angle<br />
360<br />
60 _ 120<br />
3<br />
360<br />
90 _ 90<br />
4<br />
360<br />
108 _ 72<br />
5<br />
360<br />
120 _ 60<br />
6<br />
360<br />
128.57 _ 51.43<br />
7<br />
360<br />
135 _ 45<br />
8<br />
360<br />
140 _ 40<br />
9<br />
360<br />
144 _ 36<br />
10<br />
360<br />
165 _ 15<br />
24<br />
180(n 2) _<br />
n 360 _<br />
n
2 18,000 180(n 2)<br />
n 2 100<br />
n 102<br />
3 Diagonals n 3<br />
a 0 b 1 c 2<br />
d 4 e 8<br />
4 Triangles n 2<br />
a 7 b 10 c 15 d 98<br />
5 180n<br />
a 720 b 900 c 1,160 d 3,420<br />
6 sum _<br />
180<br />
a 1,000 sides b 100 sides<br />
7 360 _<br />
n<br />
a 40 b 36 c 10 d 5<br />
360<br />
8 __<br />
# of degrees<br />
a 12 b 10 c 6 d 8<br />
180(n 2)<br />
9 # of degrees _<br />
n<br />
a 18 b 360 c 8 d 6<br />
10 sum 180(n 2)<br />
a 12 b 24 c 50 d 1,000<br />
11 Each exterior angle is 30. 360 _ 12 sides<br />
30<br />
12 180(n 2) 720<br />
180n 360 720<br />
180n 1,080<br />
n 6<br />
180(n 2)<br />
13 _ 8 360<br />
n<br />
_<br />
n<br />
180(n 2) 2,880<br />
n 2 16<br />
n 18<br />
14 m1 70, m2 75, m3 105,<br />
m4 145<br />
15 m1 90, m2 110, m3 70,<br />
m4 35, m5 95, m6 85<br />
16 m1 m2 m6 120<br />
m3 m4 m5 60<br />
17 180(n 2) 5(360)<br />
n 12<br />
18 (5x 10) (6x 25) (6x 25) <br />
(5x 10) (3x 5) 180(3)<br />
25x 65 540<br />
x 19<br />
Interior angles: 105, 139, 139, 105, 52<br />
19 x + 3x + 4x + 4x + 6x 360<br />
18x 360<br />
x 20<br />
Exterior angles: 20, 60, 80, 80, 120<br />
Interior angles: 160, 120, 100, 100, 60<br />
20 (3x 4) (7x 7) (6x 5) (5x 8) <br />
(3x 2) 5x 360<br />
29x 348<br />
x 12<br />
Exterior angles: 40, 91, 67, 68, 34, 60<br />
Interior angles: 140, 89, 113, 112, 146, 120<br />
Chapter Review (pages 212–216)<br />
1 a no slope<br />
b 1 _<br />
4<br />
c 15 _<br />
7<br />
d 5 _<br />
2<br />
e 3 _<br />
5<br />
f 0<br />
2 a 2<br />
b 1<br />
c no slope<br />
d 0<br />
e 7 _<br />
5<br />
f no slope<br />
3<br />
___ 3 m AC _ 5<br />
___ , m BC <br />
5 _ . Slopes are negative<br />
3<br />
reciprocals.<br />
___ 4 m SL m ___<br />
BC m ___<br />
PS m ___ 1. Opposite sides<br />
LA<br />
are parallel (slopes are equal) and slopes of<br />
consecutive sides are negative<br />
reciprocals (sides are perpendicular).<br />
___ 5 m PL m ___<br />
AN 1 and m ___<br />
PS m ___ 4.<br />
LA<br />
Opposite sides are parallel (slopes are<br />
equal).<br />
6 For parallel lines m and n cut by transversal<br />
a, m6 m10. So 2y 3y 10, and<br />
y 38.<br />
m1 m6 m9 m14 76<br />
m2 m5 m10 m13 104<br />
For parallel lines m and n cut by transversal<br />
b, m4 m12. So 2y 15 3y 7,<br />
and y 22.<br />
m4 m7 m12 m15 59<br />
m3 m8 m11 m16 121<br />
7 m1 42, m2 48, m3 42, m4 42<br />
8 mx 85<br />
9 mx 60<br />
10 mx 68<br />
11 130<br />
Chapter Review 53
12 a mx 45, my 45<br />
b mx 98, my 82<br />
c mx 60, my 70<br />
d mx 65, my 52<br />
e mx 67, my 78<br />
f mx 15, my 55<br />
g mx 55, my 62.5<br />
13 a 5 b 20 c 35<br />
d 594 e<br />
n 3 _<br />
2<br />
14 102 sides<br />
15 8 sides<br />
16 14 sides<br />
17 6 sides<br />
18 a mx 85<br />
b my 55<br />
19 mD 45<br />
20 m1 m2 25<br />
21 m1 140<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
22 1. 6 4<br />
2. x y<br />
3. x z<br />
4. y z (Transitive postulate)<br />
23 1. 2 4<br />
2. 1 3<br />
3. 1 2 3 4<br />
4. m r (Corresponding angles are<br />
congruent.)<br />
24 1. ___<br />
QR ___<br />
PS<br />
2. QPS QRS<br />
3. RPS QRP<br />
4. QPS RPS QRS QRP<br />
or QRP SRP<br />
5. ___<br />
QP ___<br />
RS (Corresponding angles are<br />
congruent.)<br />
25 1. ___<br />
BC bisects ABD.<br />
2. 2 3<br />
3. 1 2<br />
4. 1 3 (Transitive postulate)<br />
5. 5 1<br />
54 Chapter 9: Parallel Lines<br />
6. 3 5<br />
7. a b (Alternate interior angles are<br />
congruent.)<br />
26 1. ___<br />
AC bisects BCD.<br />
2. 3 4<br />
3. ___<br />
AB ___<br />
BC<br />
4. ABC is a right angle.<br />
5. ___<br />
CD ___<br />
AD<br />
6. CDA is a right angle.<br />
7. ___<br />
AC ___<br />
AC<br />
8. I II (Hypotenuse–acute<br />
angle)<br />
27 1. ABCD <br />
2. ___<br />
CF ___<br />
DE<br />
3. ACF BDE<br />
4. ___<br />
CF ___<br />
DE<br />
5. ___<br />
AB ___<br />
CD<br />
6. ___<br />
AB ___<br />
BC ___<br />
CD ___<br />
BC<br />
7. ___<br />
AC ___<br />
BD<br />
8. ACF BDE (SAS SAS)<br />
9. ___<br />
AF ___<br />
BE<br />
28 1. ___<br />
BE bisects ABC.<br />
2. DBE EBA<br />
3. ___<br />
DE ___<br />
BA<br />
4. DEB EBA (Alternate interior<br />
angles are congruent.)<br />
5. DEB DBE (Transitive postulate)<br />
6. ___<br />
DB ___<br />
DE<br />
7. BDE is isosceles. (Definition of<br />
isosceles triangle)<br />
29 1. ___<br />
AF<br />
2. ___<br />
AB ___<br />
CD<br />
3. ___<br />
AC ___<br />
EF<br />
4. ___<br />
AC ___<br />
CE ___<br />
EF ___<br />
CE<br />
5. ___<br />
AE ___<br />
CF<br />
6. ___<br />
AB ___<br />
CD<br />
7. BAE DCF (Corresponding angles<br />
are congruent.)<br />
8. BAE DCF (SAS SAS)<br />
9. ___<br />
BE ___<br />
DF (CPCTC)<br />
30 1. ___<br />
AB ___<br />
CB<br />
2. A C<br />
3. B B<br />
4. ABE CBD (ASA ASA)<br />
5. ___<br />
AE ___<br />
CD<br />
31 1. ___<br />
BA ___<br />
AE<br />
2. BAE is a right angle.<br />
3. 1 2<br />
4. B D
5. ___<br />
AE ___<br />
AE<br />
6. ABE EDA (AAS AAS)<br />
7. BAE DEA (CPCTC)<br />
8. DAE is a right angle.<br />
9. ___<br />
DE ___<br />
AE (Perpendicular lines<br />
intersect to <strong>for</strong>m right<br />
32 1.<br />
angles.)<br />
___<br />
BA ___<br />
DE<br />
2. CAE CEA<br />
3. ___<br />
AE ___<br />
AE<br />
4. ABE EDA (SAS SAS)<br />
5. BEA DAE<br />
33 1. 1 2<br />
2. 3 4<br />
(CPCTC)<br />
3. 5 6 (Linear pairs of<br />
congruent angles<br />
4.<br />
are congruent.)<br />
___<br />
AD ___<br />
EC<br />
5. ___<br />
DE ___<br />
DE<br />
6. ___<br />
AE ___<br />
DC (Addition postulate)<br />
7. ABE CBD (ASA ASA)<br />
8. ABE CBD<br />
34 1.<br />
(CPCTC)<br />
___<br />
RS ___<br />
ST ___<br />
TR<br />
2. 1 2<br />
3. ___<br />
RP ___<br />
TP (Converse of isosceles<br />
4.<br />
triangle <strong>the</strong>orem)<br />
___<br />
SP ___<br />
SP<br />
5. RPS TPS (SSS SSS)<br />
6. RSP TSP<br />
7.<br />
(CPCTC)<br />
___<br />
SP bisects RST. (Definition of an<br />
angle bisector)<br />
35 1. ___<br />
PT ___<br />
QR , ___<br />
RS ___<br />
PQ<br />
2. RSP and PTR (Definition of right<br />
are right angles. angles)<br />
3. RSP and PTR are right triangles.<br />
4. ___<br />
PT ___<br />
RS<br />
5. ___<br />
PR ___<br />
PR<br />
6. RSP PTR (HL HL)<br />
7. SRP TRP (CPCTC)<br />
8. PQR is isosceles. (Converse of<br />
isosceles triangle<br />
<strong>the</strong>orem)<br />
36 1. ___<br />
AS ___<br />
PR , ___<br />
BT ___<br />
PR<br />
2. ASP and BTR are right angles.<br />
3. PAS and RBT are right triangles.<br />
4. 1 2<br />
5. PAS RBT (Linear pairs of<br />
congruent angles<br />
are congruent.)<br />
6. ___<br />
AS ___<br />
BT<br />
7. PAS RBT (Leg–acute angle)<br />
8. APS BRT (CPCTC)<br />
9. PQR is isosceles. (Definition of an<br />
isosceles triangle)<br />
37 True<br />
38 True<br />
39 False<br />
40 True<br />
41 True<br />
42 False<br />
43 True<br />
44 False<br />
45 True<br />
46 False<br />
47 False<br />
48 True<br />
49 True<br />
50 True<br />
Chapter Review 55
Quadrilaterals<br />
10-2 The Parallelogram<br />
(pages 221–223)<br />
1 mP 48, mQ mS 132<br />
2 mR 30, mQ mS 150<br />
3 mQ mR 180<br />
(3x 5) (4x 25) 180<br />
7 x 210<br />
x 30<br />
mS mQ 85<br />
mP mR 95<br />
4 mP mQ 180<br />
(7x 12) (2x 3) 180<br />
9 x 189<br />
x 21<br />
mR 135<br />
mS 45<br />
5 3x 5 1 _<br />
2 (5x 3)<br />
6x 1 0 5x 3<br />
x 7<br />
PR 5(7) 3 32<br />
QS 3(7) 5 16<br />
6 PRS PRQ; QPS SRP; QMP <br />
SMR; QMR SMP<br />
7 x (5x 6) 180<br />
x 31<br />
The angles are 31 and 149.<br />
8 (3x 24) x 180<br />
x 39<br />
The angles are 39 and 141.<br />
9 x 3<br />
10 4<br />
11 mETS mBTE mTEB 42<br />
56 Chapter 10: Quadrilaterals<br />
CHAPTER<br />
10<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
12 1. ABCD and DAPQ are parallelograms.<br />
2. −−−<br />
AD ___<br />
BC (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
3. −−−<br />
AD −−<br />
PQ (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
4. −−<br />
BC −−<br />
PQ (Transitive property)<br />
13 1. ABCD is a parallelogram.<br />
2. −−<br />
BC −−−<br />
AD<br />
3. GBF GDE<br />
4. BGF DGE<br />
5. BG −−−<br />
GD (Diagonals of a parallelogram<br />
bisect<br />
each o<strong>the</strong>r.)<br />
6. BFG DEG (ASA ASA)<br />
7. −−<br />
FG −−<br />
GE<br />
8. G is <strong>the</strong> midpoint ( Definition of<br />
of −−<br />
FE . midpoint)<br />
14 1. −−−<br />
MX −−<br />
QS<br />
2. −−<br />
TX −−−<br />
QM<br />
3. QMXT is a parallelogram.<br />
4. M is <strong>the</strong> midpoint of −−−<br />
QR .<br />
5. QM −−−<br />
MR<br />
6. QM −−<br />
TX (Opposite sides of<br />
a parallelogram are<br />
congruent.)
7. −−−<br />
MR −−<br />
TX (Transitive<br />
postulate)<br />
8. XTS MXT (Alternate<br />
interior angles)<br />
9. MXT MQT (Opposite<br />
angles)<br />
10. MQT RMX (Corresponding<br />
angles)<br />
11. XTS RMX (Transitive<br />
postulate)<br />
12. XST RXM (Corresponding<br />
angles are<br />
congruent)<br />
13. MRX TXS (AAS AAS)<br />
15 1. Parallelogram ABCD<br />
2. X is <strong>the</strong> midpoint of −−<br />
BC .<br />
3. XC 1 _ BC ( Definition of a<br />
2<br />
midpoint)<br />
4. Y is <strong>the</strong> midpoint of −−−<br />
AD .<br />
5. AY 1 _ AD<br />
2<br />
6. −−−<br />
AD −−<br />
BC<br />
7. AD BC<br />
8. 1 _ AD <br />
2 1 _ BC<br />
2<br />
9. AY XC<br />
10. −−<br />
AY −−<br />
XC<br />
11. AMY CMX (Vertical angles<br />
are congruent.)<br />
12. MAY MCX (Alternate interior<br />
angles are<br />
congruent.)<br />
13. MAY MCX (AAS AAS)<br />
14. −−−<br />
XM −−−<br />
YM (CPCTC)<br />
15. a M is <strong>the</strong> midpoint (Definition of<br />
of −−<br />
XY . midpoint)<br />
16. −−−<br />
AM −−−<br />
MC (CPCTC)<br />
17. b M is <strong>the</strong> midpoint (Definition of<br />
of −−<br />
AC . midpoint)<br />
16 1. −−<br />
AC is a diagonal in parallelogram ABCD.<br />
2. −−<br />
AF −−<br />
CE<br />
3. −−<br />
FE −−<br />
FE<br />
4. −−<br />
AF −−<br />
FE −−<br />
CE −−<br />
FE<br />
5. −−<br />
AE −−<br />
CF<br />
6. DAC BCD (Definition of a<br />
parallelogram)<br />
7. BAF DCE (Alternate interior<br />
angles are<br />
congruent.)<br />
8. EAD FBC (Subtraction<br />
postulate)<br />
9. −−−<br />
AD −−<br />
BC (Opposite sides of a<br />
parallelogram are<br />
congruent.)<br />
10. ADE BCF (SAS SAS)<br />
11. AED CFB (CPCTC)<br />
12. −−<br />
DE −−<br />
BF (Alternate interior<br />
angles are congruent.)<br />
17 1. ABCD is a parallelogram.<br />
2. −−<br />
DE −−<br />
AF<br />
3. −−<br />
CF −−<br />
AF<br />
4. DAE and CFB are right angles.<br />
5. DAE CFB<br />
6. −−<br />
CA −−−<br />
AD<br />
7. −−−−<br />
AEBF<br />
8. −−<br />
EB −−<br />
EB<br />
9. −−<br />
AE −−<br />
BF (Subtraction<br />
postulate)<br />
10. DEA CFB (SAS SAS)<br />
11. −−<br />
DE −−<br />
CF (CPCTC)<br />
18 1. Parallelogram ABCD<br />
2. H is <strong>the</strong> midpoint of −−<br />
AB .<br />
3. AH 1 _ AB<br />
2<br />
4. F is <strong>the</strong> midpoint of −−−<br />
DC .<br />
5. FC 1 _ DC<br />
2<br />
6. −−<br />
AB −−−<br />
DC (Opposite sides of a<br />
parallelogram are<br />
congruent.)<br />
7. AB DC<br />
8. 1 _ AB <br />
2 1 _ DC<br />
2<br />
9. AH FC<br />
10. −−−<br />
AH −−<br />
FC<br />
11. −−−<br />
HG −−<br />
AC , −−<br />
FE −−<br />
AC<br />
12. HGA and FEC are right angles.<br />
13. HFA FEC<br />
14. −−<br />
AB −−−<br />
DC<br />
15. CAB ACD<br />
16. GAH ECF (AAS AAS)<br />
17. −−−<br />
HG −−<br />
FE<br />
19 1. Parallelogram ABCD<br />
2. −−<br />
AR −−−<br />
CM<br />
3. −−<br />
AR −−−<br />
MR −−−<br />
CM −−−<br />
MR<br />
4. −−−<br />
AM −−<br />
CR<br />
5. −−−<br />
AD −−<br />
BC<br />
6. −−−<br />
AD −−<br />
BC<br />
10-2 The Parralallogram 57
7. MAD RCB<br />
8. MAD RCB (SAS SAS)<br />
9. −−<br />
BR −−−<br />
DM (CPCTC)<br />
20 1. Parallelogram ABCD<br />
2. −−−<br />
QD bisects D.<br />
3. mCDQ 1 _ mCDP<br />
2<br />
4. −−<br />
PB bisects B.<br />
5. mABP 1 _ mABQ<br />
2<br />
6. CDP ABQ<br />
7. mCDP mABQ<br />
8. 1 _ mCDP <br />
2 1 _ mABQ<br />
2<br />
9. mCDQ mABP<br />
10. CDQ ABP<br />
11. DCQ BAP<br />
12. ABP CDQ (SAS SAS)<br />
13. a −−<br />
AP −−−<br />
CQ (CPCTC)<br />
14. −−<br />
BC −−−<br />
AD<br />
15. b −−<br />
BQ −−<br />
PD (Subtraction<br />
postulate)<br />
21 1. Parallelogram PQRS<br />
2. PS PQ<br />
3. mx mQSP<br />
4. −−−<br />
QR −−<br />
PS<br />
5. y QSP (Alternate<br />
interior angles<br />
are congruent.)<br />
6. my mQSP<br />
7. mx > my (Substitution<br />
postulate)<br />
22 1. Parallelogram MARC<br />
2. AR MA<br />
3. mAMR mARM<br />
4. ARM CMR (Alternate<br />
interior angles<br />
are congruent.)<br />
5. mARM mCMR<br />
6. mAMR mCMR (Substitution<br />
postulate)<br />
7. AMR is not congruent to CMR.<br />
58 Chapter 10: Quadrilaterals<br />
10-3 Proving That a<br />
Quadrilateral Is a<br />
Parallelogram<br />
(pages 224–225)<br />
1 Check students’ answers. The following is<br />
one possible solution.<br />
Slope of −−<br />
AB slope of −−−<br />
CD 3. Slope of<br />
−−−<br />
AD slope of −−<br />
BC 0. Both pairs of opposite<br />
sides are parallel.<br />
2 a PR 5<br />
b ( 7 _ , 8) or (3.5, 8)<br />
2<br />
3 a Check students’ answers. The following is<br />
one possible solution.<br />
DR AB 10. Slope of −−−<br />
DR slope<br />
of −−<br />
AB 0. One pair of opposite sides are<br />
both congruent and parallel.<br />
b The length of <strong>the</strong> altitude from B to −−−<br />
DR<br />
is 5.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
4 1. Parallelogram LOVE<br />
2. AOV BEL<br />
3. OAV EBL<br />
4. −−−<br />
OV −−<br />
EL<br />
5. OAV EBL (AAS AAS)<br />
5 BF 1 _ CE (segment connecting midpoints of<br />
2<br />
sides of a triangle) and CD 1 _ CE (defin-<br />
2<br />
tion of midpoint). BF CD. DF 1 _ AC and<br />
2<br />
BC 1 _ AC. BF CD and DF BC. Opposite<br />
2<br />
sides of a parallelogram have equal measure<br />
so <strong>the</strong>y are congruent.<br />
6 1. −−<br />
KL and −−<br />
EU bisect each o<strong>the</strong>r at M.<br />
2. −−−<br />
KM −−−<br />
LM<br />
3. −−−<br />
EM −−−<br />
MU<br />
4. EMK UML<br />
5. EMK UML (SAS SAS)<br />
6. −−<br />
EK −−<br />
LU (CPCTC)<br />
7. K is <strong>the</strong> midpoint of −−<br />
JE .
8. EK 1 _ EJ<br />
2<br />
9. L is <strong>the</strong> midpoint of −−−<br />
UN .<br />
10. UL 1 _ UN<br />
2<br />
11. EK UL<br />
12. 1 _ EJ <br />
2 1 _ UN<br />
2<br />
13. EJ UN<br />
14. −−<br />
EJ −−−<br />
UN<br />
15. −−<br />
EU −−<br />
EU<br />
16. LUM KEM<br />
17. EUN UEJ<br />
18. −−<br />
EN −−<br />
UJ<br />
19. JUNE is a ( Both pairs<br />
parallelogram. of opposite sides<br />
are congruent.)<br />
7 1. ABCD is a parallelogram.<br />
2. −−<br />
RC<br />
3. −−−<br />
DQ −−<br />
RC , −−<br />
AR −−<br />
RC<br />
4. DQC and ARB are right angles.<br />
5. DQC ARB<br />
6. RBA QCD (Corresponding<br />
angles are<br />
congruent.)<br />
7. −−<br />
AB −−−<br />
DC (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
8. AB DC<br />
9. ARB DQC (AAS AAS)<br />
10. −−<br />
AR −−−<br />
DQ (CPCTC)<br />
11. AR DQ<br />
12. −−<br />
AR −−−<br />
DQ Segments perpendicular<br />
to <strong>the</strong> same<br />
segment are<br />
parallel.)<br />
13. ARQD is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
parallel.)<br />
8 1. −−<br />
PE bisects −−<br />
HL at M.<br />
2. −−−<br />
HM −−−<br />
ML<br />
3. EPL PEH<br />
4. HME LMP<br />
5. HME LMP (AAS AAS)<br />
6. −−<br />
HE −−<br />
LP<br />
7. −−<br />
HE −−<br />
LP<br />
8. HELP is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
parallel.)<br />
9 1. Parallelogram ABCD<br />
2. −−−<br />
AD −−<br />
BC<br />
3. −−−<br />
AM −−−<br />
NC<br />
4. M is midpoint of −−−<br />
AD .<br />
5. AM 1 _ AD<br />
2<br />
6. N is <strong>the</strong> midpoint of −−<br />
BC .<br />
7. NC 1 _ BC<br />
2<br />
8. −−−<br />
AD −−<br />
BC<br />
9. AD BC<br />
10. 1 _ AD <br />
2 1 _ BC<br />
2<br />
11. AM NC<br />
12. −−−<br />
AM −−−<br />
NC<br />
13. ANCM is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
10 1. BR and DM <br />
2. 2 3<br />
3.<br />
parallel.)<br />
−−<br />
BC −−−<br />
AD<br />
4. 1 4<br />
5. BAD DCB (Supplements of<br />
congruent angles<br />
6.<br />
are congruent.)<br />
−−<br />
BD −−<br />
BD<br />
7. BCD DAB<br />
8.<br />
(AAS AAS)<br />
−−<br />
BC −−−<br />
AD<br />
9. ABCD is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
parallel.)<br />
11 1. −−<br />
QS bisects −−<br />
PR (at M).<br />
2. −−−<br />
QM −−−<br />
MS<br />
3. 1 2<br />
4. QMR SMP<br />
5. QMR SMP<br />
6.<br />
(AAS AAS)<br />
−−−<br />
QR −−<br />
SP<br />
7. −−<br />
PR −−<br />
PR<br />
8. PSR RQP<br />
9.<br />
(SAS SAS)<br />
−−<br />
QP −−<br />
SP<br />
10. PQRS is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
12 1.<br />
parallel.)<br />
−−−<br />
QV bisects −−<br />
RT .<br />
2. −−<br />
RS −−<br />
ST<br />
3. −−−<br />
QR −−<br />
PV<br />
4. RQS TVS (Alternate interior<br />
angles)<br />
10-3 Proving That a Quadrilateral Is a Paralallogram 59
5. RSQ TSV (Vertical angles)<br />
6. QRS VTS (AAS AAS)<br />
7. −−−<br />
QR −−<br />
PT (CPCTC)<br />
8. −−<br />
PT −−<br />
TV<br />
9. −−−<br />
QR −−<br />
PT<br />
10. −−−<br />
QR −−<br />
PT<br />
11. PQRT is a ( One pair of opparallelogram.<br />
posite sides is both<br />
congruent and<br />
parallel.)<br />
13 1. Parallelogram ABCD<br />
2. −−−<br />
DC −−<br />
AB (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
3. −−<br />
DF −−<br />
BE<br />
4. −−−−<br />
DFEB<br />
5. CDF ABE (Alternate interior<br />
angles are congruent.)<br />
6. DFC BEA (SAS SAS)<br />
7. −−<br />
CF −−<br />
AE (CPCTC)<br />
8. −−−<br />
AD −−<br />
CB (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
9. ADF EBA<br />
10. AFD CEB (SAS SAS)<br />
11. −−<br />
AF −−<br />
CE (CPCTC)<br />
12. AECF is a (Both pairs<br />
parallelogram. of opposite sides are<br />
congruent.)<br />
14 1. −−<br />
KJ is a diagonal in parallelogram KBJD.<br />
2. −−<br />
KA −−<br />
JC<br />
3. BJC <strong>AK</strong>D<br />
4. −−<br />
BJ −−−<br />
KD<br />
5. BJC KAD (SAS SAS)<br />
6. −−<br />
BC −−−<br />
AD (CPCTC)<br />
7. −−<br />
BK −−<br />
JD<br />
8. CJD BKA<br />
9. ABK CDJ (SAS SAS)<br />
10. −−<br />
AB −−−<br />
CD (CPCTC)<br />
11. ABCD is a ( Both pairs<br />
parallelogram. of opposite sides are<br />
congruent.)<br />
15 1. −−<br />
BD is a diagonal in parallelogram<br />
ABCD.<br />
2. BEC AFD<br />
3. EBC ADF (Alternate interior<br />
angles are congruent.)<br />
4. −−<br />
BC AD (Opposites sides of<br />
a parallelogram are<br />
congruent.)<br />
60 Chapter 10: Quadrilaterals<br />
5. CBE AFD<br />
6.<br />
(AAS AAS)<br />
−−<br />
AF −−<br />
EC<br />
7.<br />
(CPCTC)<br />
−−<br />
DF −−<br />
EB<br />
8. ABE FDC<br />
9.<br />
(CPCTC)<br />
−−<br />
AB −−−<br />
DC<br />
10. ABE CDE<br />
11.<br />
(SAS SAS)<br />
−−<br />
EA −−<br />
CF (CPCTC)<br />
10-4 Rectangles<br />
(pages 227–228)<br />
1 (1) are congruent<br />
2 AR DR<br />
3(4x 3) 10x 1<br />
x 5<br />
AR CR DR BR 51<br />
3 AR BR<br />
2(x 6) 3x 20<br />
x 8<br />
AR CR 4<br />
BD 8<br />
4 DR CR<br />
4(3x 10) 3(x 2) 12<br />
x 46 _<br />
9<br />
AR 46 _<br />
9<br />
AC BD 128 _<br />
9<br />
5 AC BD<br />
3(2x 5) 1 _ (4x 4) <br />
4 2 _ (12x 3) 5x<br />
3<br />
x 2<br />
AC 24<br />
DR 12<br />
6 2x 3 0 36<br />
x 3<br />
7 PR QS<br />
4x 3 6x 7<br />
x 5<br />
PR 4(5) 3 23<br />
QS 6(5) 7 23<br />
√ PR QS √ 23 23 23<br />
8 mADB mDAC 90 49 41<br />
9 (5, 6)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
10 1. Rectangle ABCD<br />
2. ABCD is a parallelogram.<br />
3. −−<br />
BC −−−<br />
AD<br />
4. −−<br />
AB −−−<br />
CD (Opposite sides of<br />
a parallelogram are<br />
5.<br />
congruent.)<br />
−−<br />
BD −−<br />
CA (Diagonals of a<br />
rectangle are<br />
congruent.)<br />
6. CDA BAD (SSS SSS)<br />
7. CAD BDA<br />
11 1. Rectangle PQRS<br />
(CPCTC)<br />
2. PQRS is a parallelogram.<br />
3. −−<br />
PQ −−<br />
PQ<br />
4. −−−<br />
QR −−<br />
PS<br />
5. SQP PQR (SSS SSS)<br />
6. 1 2 (CPCTC)<br />
12 1. Rectangle ABCD<br />
2. ABP and NCD are right angles.<br />
3. ABP and NCD are right triangles.<br />
4. −−<br />
AP −−−<br />
DN<br />
5. −−−<br />
DC −−<br />
AB (Opposite sides of<br />
a parallelogram are<br />
congruent.)<br />
6. ABP DCN (HL HL)<br />
7. DNC APB (CPCTC)<br />
8. PAD APB<br />
9. NDA APB<br />
(Alternate interior<br />
angles are<br />
congruent.)<br />
10. PAD NDA (Transitive<br />
11.<br />
postulate)<br />
−−<br />
AE −−<br />
DE (Definition of an<br />
isosceles triangle)<br />
13 By <strong>the</strong> addition postulate of inequality, −−<br />
AY<br />
is not congruent to −−<br />
TX , and AMY is not<br />
congruent to THX. There<strong>for</strong>e, −−−<br />
MY is not<br />
congruent to −−−<br />
HX .<br />
14 1. Rectangle ABCD<br />
2. N is <strong>the</strong> midpoint of −−−<br />
CD .<br />
3. CN DN<br />
4. −−−<br />
CN −−−<br />
DN<br />
5. BCN NDA (A rectangle is<br />
6.<br />
equiangular.)<br />
−−<br />
BC −−−<br />
AD (Opposite sides of<br />
a parallelogram are<br />
7.<br />
congruent.)<br />
−−<br />
BN −−−<br />
AN (CPCTC)<br />
15 1. −−<br />
AB −−−<br />
CD<br />
2. BJH and AJH are linear angles.<br />
3. mBJH mAJH 180<br />
4. 1 _ mBJH <br />
2 1 _ mAJH 90<br />
2<br />
(Division postulate)<br />
5. −−<br />
JG bisects BJH.<br />
6. mHJG 1 _ mBJH<br />
2<br />
7. −−<br />
EJ bisects AJH.<br />
8. mEJH 1 _ mAJH<br />
2<br />
9. mHJG mEJH 90 ( Substitution)<br />
10. mEJG 90<br />
11. EJG is a right angle.<br />
12. CHJ BJH<br />
13. EHJ GJH<br />
14. AJH DHJ<br />
15. HJE JHG<br />
16. −−<br />
HJ −−<br />
HJ<br />
17. EJH GHJ<br />
18. −−<br />
EJ −−−<br />
GH<br />
19. −−<br />
EH −−<br />
JG<br />
20. EJGH is a parallelogram.<br />
21. EJGH is a rectangle. (Definition of a<br />
rectangle)<br />
16 1. Rectangle PQRS<br />
2. −−<br />
PA −−<br />
CS<br />
3. −−<br />
PA −−<br />
AC −−<br />
CS −−<br />
AC<br />
4. −−<br />
PC −−<br />
AS<br />
5. −−<br />
QP −−<br />
RS<br />
6. QPS RSP<br />
7. QPC RSA (SAS SAS)<br />
8. a 1 2 (CPCTC)<br />
9. PQR and SRQ are right angles.<br />
10. 3 is complementary to 1.<br />
11. 4 is complementary to 2.<br />
12. b 3 4 (Complements of<br />
congruent angles<br />
are congruent.)<br />
13. c −−<br />
QB −−<br />
RB (Definition of an<br />
isosceles triangle)<br />
10-5 Rhombuses<br />
(pages 229–231)<br />
1 mB mD 105; mC 75.<br />
AB CD 7<br />
2 mADB 66<br />
10-5 Rhombuses 61
3 4x 2 3x 3<br />
x 5<br />
RS 18<br />
4 Perimeter of MABC 8<br />
5 mADC 110<br />
6 2x 2 x 8<br />
x 10<br />
CD 18<br />
7 mADC 94<br />
8 a Midpoint of −−<br />
AC midpoint of −−<br />
BD (6, 5)<br />
b Slope of −−<br />
AC 1, slope of −−<br />
BD 1.<br />
Slopes are negative reciprocals, diagonals<br />
are perpendicular.<br />
9 a (9, 5)<br />
b PQ PS 5 √ 2<br />
c Slope of −−<br />
PR 1 _ , slope of<br />
3 −−<br />
QS 3.<br />
Slopes are negative reciprocals, diagonals<br />
are perpendicular.<br />
10 a Slope of −−−<br />
AD slope of −−<br />
BC k _<br />
x<br />
.<br />
AD BC √ x 2 k 2 . ABCD is a parallelogram.<br />
(One pair of opposite sides have<br />
<strong>the</strong> same length and are parallel.)<br />
b Slope of −−<br />
CA k _<br />
−−<br />
, slope of BD <br />
x k k _<br />
x k ,<br />
not negative reciprocals. The diagonals are<br />
not perpendicular.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
11 −−−<br />
QR −−<br />
AC , −−<br />
PS −−<br />
AC , −−−<br />
QR −−<br />
PS . Likewise,<br />
−−<br />
QP −−<br />
BD , −−<br />
RS −−<br />
BD , and −−<br />
QP −−<br />
RS . Thus, PQRS<br />
is a parallelogram. Since −−<br />
AC −−<br />
BD , −−−<br />
QR<br />
and −−<br />
PS are perpendicular to −−<br />
QP and −−<br />
RS .<br />
Thus, PQRS has at least one right angle, and<br />
PQRS is a rectangle.<br />
12 1. Rhombus ABCD<br />
2. −−<br />
ED<br />
3. −−<br />
BF −−<br />
AC<br />
4. −−<br />
EF −−<br />
AC<br />
5. F bisects −−<br />
AC .<br />
6. EA EC<br />
62 Chapter 10: Quadrilaterals<br />
7. −−<br />
EA −−<br />
EC<br />
8. ACE is isosceles. (Definition of an<br />
isosceles triangle)<br />
13 Prove that −−<br />
DF −−<br />
AB and −−<br />
DE −−<br />
BF by using<br />
<strong>the</strong> midpoints and alternate interior angles.<br />
Thus, EBFD is a parallelogram. By <strong>the</strong> division<br />
postulate, −−<br />
EB −−<br />
FB . EBFD is a rhombus<br />
because a rhombus is a parallelogram with<br />
two congruent consecutive sides.<br />
14 1. Parallelogram ABCD<br />
2. −−<br />
AB −−−<br />
CD<br />
3. BAD 2<br />
4. mBAD m2<br />
(Corresponding angles<br />
are congruent.)<br />
5. BAD 1 CAD<br />
6. mBAD m1 (Whole is greater<br />
than a part.)<br />
7. m2 m1 (Substitution<br />
postulate)<br />
15 1. Rhombus PQRS<br />
2. −−<br />
QS −−<br />
PR<br />
3. −−−<br />
QC −−<br />
CS<br />
4. B is <strong>the</strong> midpoint of −−−<br />
QC .<br />
5. D is <strong>the</strong> midpoint of −−<br />
CS .<br />
6. −−<br />
BC −−−<br />
CD<br />
7. ___<br />
AC ___<br />
AC<br />
8. ABC ADC<br />
9.<br />
(SAS SAS)<br />
−−<br />
AB −−−<br />
AD<br />
(CPCTC)<br />
10. BAD is isosceles. (Definition of an<br />
isosceles triangle)<br />
16 If a quadrilateral is equilateral, it is a parallelogram<br />
with a pair of consecutive congruent<br />
sides.<br />
17 The diagonal creates two congruent triangles<br />
(ASA ASA) and corresponding congruent<br />
sides are consecutive.<br />
18 1. −−−<br />
CD BE <br />
2. −−−<br />
CD −−<br />
BA<br />
3. −−−<br />
AD BF <br />
4. −−−<br />
AD −−<br />
BC<br />
5. ABCD is a ( Definition of a<br />
parallelogram. parallelogram)<br />
6. BAD BCD (Opposite angles<br />
7. BG bisects FBE.<br />
are congruent.)<br />
8. CBD ABD<br />
9. ABCD is a rhombus. (Diagonal −−<br />
BD<br />
bisects opposite<br />
angles.)
10-6 Squares<br />
(pages 232–233)<br />
1 (4) A rectangle is a square.<br />
2 (3) sides and angles are congruent<br />
3 (4) A trapezoid is a parallelogram.<br />
4 (3) −−<br />
AC −−−<br />
DC<br />
5 (1) congruent and bisect <strong>the</strong> angles to which<br />
<strong>the</strong>y are drawn<br />
6 (2) x √ 2<br />
7 Slope of −−−<br />
DA slope of −−<br />
VE 3 _ . Slope of<br />
4 −−<br />
ED slope of −−<br />
AV 4 _ . A parallelogram<br />
3<br />
has two pairs of opposite sides that are parallel.<br />
A rectangle is a parallelogram in which<br />
consecutive sides have slopes that are negative<br />
reciprocals. Slope of −−−<br />
DV 1 _ . Slope<br />
7<br />
of −−<br />
AE 7. Slopes of <strong>the</strong> diagonals are negative<br />
reciprocals, thus <strong>the</strong>y are perpendicular.<br />
There<strong>for</strong>e, DAVE is a square.<br />
8 Slope of −−−<br />
MA slope of −−<br />
TH 4 _ . Slope of<br />
3 −−−<br />
HM slope of −−<br />
AT 3 _ . A parallelogram has<br />
4<br />
two pairs of opposite sides that are parallel.<br />
A rectangle is a parallelogram in which consecutive<br />
sides have slopes that are negative<br />
reciprocals. Slope of −−−<br />
MT 1 _ . Slope of<br />
7 −−−<br />
AH 7. Slopes of <strong>the</strong> diagonals are negative<br />
reciprocals, thus <strong>the</strong>y are perpendicular.<br />
There<strong>for</strong>e, MATH is a square.<br />
9 The diagonals bisect <strong>the</strong> vertex angles, creating<br />
four congruent isosceles triangles. The<br />
bisected angles (<strong>the</strong> base angles of <strong>the</strong> triangles)<br />
measure 45. The vertex angles thus<br />
measure 90. The diagonals cross, <strong>for</strong>ming<br />
90 angles.<br />
10 a Slope of −−<br />
PQ slope of −−<br />
RS 0. −−−<br />
QR and −−<br />
SP<br />
have no slope. A parallelogram has two<br />
pairs of opposite sides that are parallel.<br />
A rectangle is a parallelogram in which<br />
consecutive sides are perpendicular.<br />
b PQ RS 20. QR PS 7. Since all<br />
sides are not congruent to each o<strong>the</strong>r,<br />
PQRS is not a square.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
11 a Given ABCD is a square. BFE is a right<br />
angle because −−<br />
EF −−<br />
BD . mEBF 45<br />
because diagonal −−<br />
BD bisects right angle<br />
ABC. mFEB 45. There<strong>for</strong>e, −−<br />
BF −−<br />
EF .<br />
b 1. ABCD is a square.<br />
2. −−<br />
EF −−<br />
BD<br />
3. EFD is a right angle.<br />
4. DAE is a right ( Definition of<br />
angle. a square)<br />
5. DAE and DFE are right triangles.<br />
6. (Draw −−<br />
DE ). −−<br />
DE −−<br />
DE<br />
7. DAE DFE (HL HL)<br />
8. −−<br />
EF −−<br />
EA (CPCTC)<br />
12 1. Rhombus ABCD with diagonals −−<br />
AC<br />
and −−<br />
BD intersecting at E.<br />
2. 1 2<br />
3. −−<br />
BE −−<br />
CE<br />
4. E bisects −−<br />
AC and −−<br />
BD .<br />
5. −−<br />
BD −−<br />
AC<br />
6. ABCD is a rectangle.<br />
7. ABCD has all right angles.<br />
8. ABCD is a square. (A square is a<br />
rhombus in which<br />
all angles are right<br />
angles.)<br />
10-7 Trapezoids<br />
(pages 236–238)<br />
1 (2) They are congruent.<br />
2 (3) ADC ABC<br />
3 a x z 70; y 110<br />
b x 73; y z 107<br />
c x 40; y 108; z 32<br />
d x 70; y 44; z 66<br />
e x 130; y 20; z 30<br />
f x 82; y z 41<br />
4 Slope of −−−<br />
MA slope of −−<br />
TH 1 _ . These legs<br />
2<br />
are parallel. Slope of −−<br />
AT 3 _ HM has<br />
4 . −−−<br />
no slope. These legs are not parallel.<br />
AT HM 10. There<strong>for</strong>e, MATH is an<br />
isosceles trapezoid.<br />
10-7 Trapezoids 63
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
5 1. ABCD is an isosceles trapezoid.<br />
2. BAD CDA<br />
3. −−<br />
BC −−−<br />
AD<br />
4. NBC BAD<br />
5. NCB CDA<br />
(Corresponding<br />
angles are<br />
congruent.)<br />
6. NBC NCB (Substitution<br />
postulate)<br />
7. NBC is isosceles. (Base angles of an<br />
isosceles triangle<br />
are congruent.)<br />
6 1. Isosceles trapezoid ABCD<br />
2. −−<br />
AB −−−<br />
DC<br />
3. BAD CDA (Base angles of an<br />
isosceles trapezoid<br />
4.<br />
are congruent.)<br />
−−−<br />
AD −−−<br />
AD<br />
5. ADB DAC<br />
7 1. Trapezoid ABCE<br />
2.<br />
(SAS SAS)<br />
−−<br />
BD −−−<br />
AD<br />
3. −−<br />
AB −−<br />
CE<br />
4. A B<br />
5. CED A<br />
6. ECD B (Corresponding<br />
angles are<br />
congruent.)<br />
7. CED ECD (Substitution<br />
8.<br />
postulate)<br />
−−−<br />
CD −−<br />
ED<br />
9. −−<br />
BD −−−<br />
CD ( Subtraction<br />
−−−<br />
AD −−<br />
ED<br />
10.<br />
postulate)<br />
−−<br />
BC −−<br />
AE<br />
11. ABCE is an (Nonparallel sides<br />
isosceles trapezoid. of an isosceles<br />
trapezoid are<br />
congruent.)<br />
8 1. Quadrilateral PQRS<br />
2. −−−<br />
QAB , −−−<br />
RAS , and −−−<br />
PSB<br />
3. −−<br />
QB bisects −−<br />
RS .<br />
4. −−<br />
RA −−<br />
AS<br />
64 Chapter 10: Quadrilaterals<br />
5. −−−<br />
PSB −−−<br />
QR<br />
6. RAQ BAS (Vertical angles are<br />
congruent.)<br />
7. QRA BSA (Alternate interior<br />
angles are<br />
congruent.)<br />
8. QRA BSA<br />
9.<br />
(ASA ASA)<br />
−−−<br />
QA −−<br />
AB<br />
9 1. Isosceles trapezoid ABCD<br />
2. A D<br />
3. −−−<br />
AD −−<br />
BC<br />
4. −−<br />
AB −−−<br />
DC<br />
5. E is <strong>the</strong> midpoint of −−−<br />
AD .<br />
6. −−<br />
AE −−<br />
ED<br />
7. ABE DCE<br />
8. a<br />
(SAS SAS)<br />
−−<br />
BE −−<br />
CE<br />
9. BE CE<br />
(CPCTC)<br />
10. BCE is isosceles. (Definition of isos-<br />
11. b<br />
celes triangle)<br />
−−<br />
EH −−<br />
BC (The median from<br />
<strong>the</strong> vertex angle of<br />
an isosceles triangle<br />
is perpendicular<br />
to <strong>the</strong> base.)<br />
10 1. Isosceles trapezoid PQRS<br />
2. −−<br />
PQ −−<br />
RS<br />
3. −−−<br />
QR −−<br />
PS<br />
4. P S<br />
5. −−<br />
PS −−<br />
PS<br />
6. PQS SRP (SAS SAS)<br />
7. PQS SRP (CPCTC)<br />
8. QAP BAS (Vertical angles are<br />
congruent.)<br />
9. PAQ SAR<br />
11 1. Trapezoid ABCD<br />
2.<br />
(AAS AAS)<br />
−−<br />
BR −−−<br />
AD<br />
3. −−−<br />
CM −−−<br />
AD<br />
4. ARB and DMC are right angles.<br />
5. ARB and DMC are right triangles.<br />
6. −−<br />
AB −−−<br />
CD<br />
7. BAR CDM<br />
8. ARB DMC (Hypotenuse–acute<br />
angle)<br />
9. 1 2 (CPCTC)<br />
12 1. Trapezoid PQRS<br />
2. Q R<br />
3. −−<br />
PA −−−<br />
QR , −−<br />
SE −−−<br />
QR<br />
4. PAQ and SER are right angles.<br />
5. PAQ and SER are right triangles.
6. AP ES (Distances between<br />
parallel lines are<br />
7.<br />
equal.)<br />
−−<br />
AP −−<br />
ES<br />
8. PAQ SER<br />
9.<br />
(Leg–Acute Angle)<br />
−−<br />
QP −−<br />
RS (CPCTC)<br />
10-8 Kites<br />
(page 239)<br />
1 A kite is a quadrilateral with only two pairs<br />
of adjacent congruent sides. A rhombus has<br />
four congruent sides.<br />
2 True<br />
3 True<br />
4 False<br />
5 True<br />
6 True<br />
7 False<br />
8 False<br />
9 True<br />
10 a −−<br />
AC<br />
b 1 8; 2 7; 3 6; 4 5<br />
11 a 90 b 45 c 17<br />
d m4 25 and mKLM 50<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
12 1. −−<br />
QS is <strong>the</strong> perpendicular bisector of −−<br />
PR .<br />
2. −−<br />
PT −−<br />
TR<br />
3. QTP QTR<br />
4. −−<br />
QT −−<br />
QT<br />
5. QPT QRT (SAS SAS)<br />
6. STP STR<br />
7. −−<br />
TS −−<br />
TS<br />
8. STP STR (SAS SAS)<br />
9. −−<br />
QP −−−<br />
QR (CPCTC)<br />
10. −−<br />
SP −−<br />
SR (CPCTC)<br />
11. −−<br />
QT −−<br />
ST<br />
12. PQRS is not ( Definition of<br />
a rhombus. a rhombus)<br />
13. PQRS is a kite. (Definition of a<br />
kite)<br />
10-9 Areas of Polygons<br />
(pages 243–245)<br />
1 4<br />
2 16<br />
3 9, 12, 15<br />
4 Length is 4 ft and width is 5 ft.<br />
5 a h 10 x<br />
b A x(10 x)<br />
6 a x 2 b 9 x 2 c x 2 4x 4<br />
d x 2 4x 4 e 4 x 2 4x 1<br />
7 a 25 _<br />
2<br />
b 18<br />
49<br />
c _<br />
2<br />
d 1<br />
8 11<br />
e 9<br />
9 a 12 b 45 c 20<br />
d 18 √ 3 e 45 _<br />
2<br />
10 √ 36 144 6 √ 5 <br />
11 a BD 10<br />
12 72 √ 3<br />
b 120<br />
13 s 6<br />
14 a 13 b 120 c 120 _<br />
13<br />
15 a 21x b 10 x 2 c 4x 12<br />
d 18x 12 e 2 x 2 16 a x 4<br />
b 4(4) 4 12<br />
17 8<br />
18 27 √ 3<br />
19 30 √ 2<br />
20 10 and 20<br />
6x<br />
21 a 96<br />
22 72<br />
23 100<br />
24 a 6<br />
b 260 c 60<br />
b SMP and AML are similar. Let x be <strong>the</strong><br />
perpendicular drawn from M to −−<br />
AP .<br />
Then x _ 20<br />
<br />
6 x _ , and x 5.<br />
4<br />
25 a _ x 10<br />
<br />
12 x _<br />
20<br />
b x 4<br />
c Area of RCS 20. Area of QCT 80.<br />
4(8 11)<br />
26 a A _<br />
2 38 b 9 ( 5, _<br />
2 )<br />
27 a Slope of −−<br />
JE 3 _ , slope of<br />
4 −−<br />
EN 4 _ . The<br />
3<br />
slopes of <strong>the</strong> two legs are negative reciprocals,<br />
<strong>the</strong>re<strong>for</strong>e perpendicular, <strong>for</strong>ming a<br />
right angle.<br />
b A 1 _ (JE)(EN) 25<br />
2<br />
10-9 Areas of Polygons 65
28 Enclose PAT in a large rectangle and subtract<br />
<strong>the</strong> excess areas.<br />
a Area of PAT 120 60 24 6 12<br />
18<br />
b AT 10<br />
c 3.6<br />
29 a SA SM √ 52 ; sides are congruent.<br />
Slope of −−<br />
SA 3 _ , slope of<br />
2 −−−<br />
SM 2 _ ;<br />
3<br />
slopes are negative reciprocals, <strong>the</strong>re<strong>for</strong>e<br />
perpendicular and <strong>for</strong>ming a right angle.<br />
b A 1 _ (SA)(SM) 26<br />
2<br />
c √ 26<br />
30 Subdivide pentagon SIMON into two triangles<br />
and a trapezoid and take <strong>the</strong> sum of all<br />
<strong>the</strong> areas.<br />
Area 72<br />
31 Divide pentagon JANET into two triangles<br />
and take <strong>the</strong> sum of <strong>the</strong> two areas.<br />
Area 64<br />
32 Enclose quadrilateral RYAN in a large rectangle<br />
and subtract <strong>the</strong> excess areas.<br />
Area 72 13 _ 6 12 12 35.5<br />
2<br />
33 −−<br />
NI and −−<br />
CK have zero slope, <strong>the</strong> two sides are<br />
parallel. −−−<br />
NK −−<br />
IC m √ 10 . There<strong>for</strong>e, NICK<br />
is an isosceles trapezoid.<br />
34 Slope of −−<br />
JO slope of −−−<br />
HN 1 _ ; <strong>the</strong> sides are<br />
3<br />
parallel. Slope of −−<br />
NJ 3. The slopes are<br />
negative reciprocals, <strong>the</strong>re<strong>for</strong>e perpendicular,<br />
<strong>for</strong>ming a right angle. Slope of −−−<br />
OH is not<br />
equal to <strong>the</strong> slope of −−<br />
NJ , <strong>the</strong>re<strong>for</strong>e, only one<br />
pair of sides are parallel. JOHN is a right<br />
trapezoid.<br />
35 a 38 b 28 c 28 d (5, 4)<br />
Chapter Review (pages 245–248)<br />
1 (4) The diagonals of a parallelogram bisect<br />
each o<strong>the</strong>r.<br />
2 (4) mB mC 360<br />
3 (1) rhombus<br />
4 (1) a rhombus<br />
5 (3) A rhombus is a square.<br />
6 (4) parallelogram<br />
7 (3) 3<br />
8 (4) −−<br />
QS and −−<br />
PR bisect each o<strong>the</strong>r.<br />
9 (1) <strong>the</strong> diagonals are congruent<br />
10 a square b rhombus<br />
c parallelogram d rectangle<br />
66 Chapter 10: Quadrilaterals<br />
11 3x 4 0 x 50<br />
x 5<br />
12 5<br />
13 5 x 90<br />
x 18<br />
14 (3x 20) (7x 4 0) 180<br />
x 20<br />
15 2<br />
16 mDAE 90 75 15<br />
17 (4, 3)<br />
18 3x 1 5 7x 55<br />
x 10<br />
19 a mACD mCAB 30<br />
b rectangle<br />
180 75<br />
20 mAEK _ 52.5<br />
2<br />
21 5 ft<br />
22 (1, 1)<br />
23 3 √ 2<br />
24 mA 45<br />
25 h 8<br />
26 Rhombus. Since <strong>the</strong> triangles are congruent,<br />
<strong>the</strong> opposite angles that are originally vertex<br />
angles are congruent. By <strong>the</strong> addition postulate,<br />
<strong>the</strong> opposite angles <strong>for</strong>med by joining<br />
<strong>the</strong> triangles at <strong>the</strong>ir bases are congruent.<br />
Thus <strong>the</strong> quadrilateral is a parallelogram.<br />
All sides are congruent. The parallelogram is<br />
a rhombus.<br />
27 PS QR<br />
2x 3 x 2<br />
x 5<br />
PS QR SR PQ 7<br />
28 a (2x 8) 3(x 3 4) 180<br />
x 14<br />
mABC mCDA 144<br />
mDAB mBCD 36<br />
b AE EC<br />
4 y 6y 36<br />
y 18<br />
BE ED<br />
3x 1 x 13<br />
x 7<br />
AC 144, BD 40<br />
29 mABP mBAP 90<br />
(5x 10) (2x 4) 90<br />
x 12<br />
mDCB 140, mAPB 90<br />
30 a mR 150 b mRAD 150<br />
c mGAD 135 d mD 30
31 a 20 √ 3 35<br />
b 20<br />
c 200 √ 3 346.4<br />
32 a AB BC 5 √ 2 . ABC is isosceles because<br />
<strong>the</strong>re is one pair of congruent sides.<br />
b Slope of −−<br />
AB 1. Slope of −−<br />
BC 1. Slopes<br />
are negative reciprocals, thus −−<br />
AB −−<br />
BC .<br />
c Area of ABC 25<br />
33 −−<br />
DE −−<br />
AV because <strong>the</strong> slope of −−<br />
DE slope<br />
of −−<br />
AV 1 _ .<br />
3 −−−<br />
DA is not parallel to −−<br />
VE because<br />
<strong>the</strong> slope of −−−<br />
DA is not equal to <strong>the</strong> slope<br />
of −−<br />
VE . DA VE 5; nonparallel sides are<br />
congruent. There<strong>for</strong>e, DAVE is an isosceles<br />
trapezoid.<br />
34 Slope of −−<br />
PQ slope of −−<br />
RS 3 _ . Slope of<br />
4 −−<br />
PS slope of −−−<br />
RQ 4 _ . Slope of<br />
3 −−<br />
PR 1 _ .<br />
7<br />
Slope of −−<br />
QS 7. PQRS is a parallelogram<br />
because opposite sides have <strong>the</strong> same slope,<br />
and a rhombus because <strong>the</strong> diagonals are<br />
perpendicular—slopes are negative reciprocals,<br />
and a square because consecutive sides<br />
are perpendicular—slopes are negative reciprocals.<br />
35 Slope of −−<br />
AB slope of −−−<br />
CD 0. Slope of<br />
−−<br />
BC slope of −−−<br />
AD 4 _ . Slope of<br />
3 −−<br />
AC 2.<br />
Slope of −−<br />
BD 1 _ . ABCD is a parallelogram<br />
2<br />
because opposite sides have <strong>the</strong> same slope,<br />
and a rhombus because <strong>the</strong> diagonals are<br />
perpendicular—slopes are negative<br />
reciprocals.<br />
36 Slope of −−<br />
LE slope of −−<br />
AF 4 _ . Slope<br />
5<br />
of −−<br />
EA slope of −−<br />
FL 4 _ . Slope of<br />
5 −−<br />
LA is<br />
udefined. Slope of −−<br />
EF 0. LEAF is a parallelogram<br />
because opposite sides have <strong>the</strong> same<br />
slope, and a rhombus because <strong>the</strong> diagonals<br />
are perpendicular.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
37 1. Quadrilateral TRIP<br />
2. −−<br />
TR −−<br />
RI<br />
3. m1 m2<br />
4. 1 2<br />
5. −−<br />
RA −−<br />
RA<br />
6. a RAT RAI (SAS SAS)<br />
7. −−<br />
TA −−<br />
AI (CPCTC)<br />
8. TAP IAP (Supplements of<br />
congruent angles<br />
are congruent.)<br />
9. −−<br />
AP −−<br />
AP<br />
10. TAP IAP (SAS SAS)<br />
11. b 3 4 (CPCTC)<br />
38 1. Parallelogram PQRS<br />
2. Diagonal −−<br />
QS bisects PQR.<br />
3. PQS RQS<br />
4. RQS QSP (Alternate interior<br />
angles are congruent.)<br />
5. PQS QSR (Alternate interior<br />
angles are congruent.)<br />
6. RQS QSR (Transitive postulate)<br />
7. QSP QSR<br />
8. −−<br />
PS −−<br />
PQ<br />
9. PS PQ<br />
10. PQRS is a rhombus. (A rhombus is a<br />
parallelogram with<br />
two congruent<br />
consecutive sides.)<br />
39 1. ABED is a rhombus.<br />
2. −−<br />
BD intersects −−−−<br />
AOEC at O.<br />
3. BOE and DOE are right angles.<br />
4. BOE and DOE are right triangles.<br />
5. −−<br />
OE −−<br />
OE<br />
6. −−<br />
BE −−<br />
DE<br />
7. BOE DOE (HL HL)<br />
8. −−<br />
BO −−−<br />
OD (CPCTC)<br />
9. BOC and DOC are right triangles.<br />
10. −−−<br />
OC −−−<br />
OC<br />
11. BOC DOC (Leg-leg)<br />
12. a −−<br />
BC −−−<br />
DC (CPCTC)<br />
13. −−<br />
AB −−−<br />
AD<br />
14. −−<br />
AC −−<br />
AC<br />
15. ABC ADC (SSS SSS)<br />
16. b ABC ADC (CPCTC)<br />
40 Mark <strong>the</strong> point of intersection of <strong>the</strong> diagonals<br />
E. The diagonals bisect each o<strong>the</strong>r,<br />
thus AE DE. By <strong>the</strong> triangle inequality<br />
<strong>the</strong>orem, mADE mDCE and<br />
mADC mDCB by <strong>the</strong> multiplication<br />
postulate of inequality.<br />
Chapter Review 67
<strong>Geometry</strong> of Three<br />
Dimensions<br />
11-1 Points, Lines, and<br />
Planes<br />
(pages 251–252)<br />
1 False<br />
2 False<br />
3 True<br />
4 True<br />
5 True<br />
6 True<br />
7 True<br />
8 False<br />
9 True<br />
10 False<br />
11 False<br />
12 H<br />
13 E<br />
14 C<br />
15 G<br />
16 C<br />
17 G<br />
18 E<br />
19 D<br />
20 a −−<br />
AB −−<br />
FG , −−<br />
AB −−−<br />
HC , −−<br />
AB −−<br />
ED ,<br />
−−−<br />
AH −−−<br />
DG , −−−<br />
AH −−<br />
CB , −−−<br />
AH −−<br />
EF , −−<br />
FA −−<br />
GB ,<br />
−−<br />
FA −−−<br />
DC , −−<br />
FA −−<br />
EH<br />
b There are many possible answers. The following<br />
are just a few.<br />
−−<br />
AB & −−<br />
EF , −−<br />
AB & −−−<br />
DG , −−<br />
BC & −−<br />
EF , −−<br />
BC &<br />
−−<br />
ED , −−−<br />
DC & −−−<br />
AH , −−−<br />
DC & −−<br />
EF .<br />
21 Infinitely many<br />
22 Infinitely many<br />
23 Infinitely many<br />
24 One<br />
25 One<br />
26 Infinitely many<br />
68 Chapter 11: <strong>Geometry</strong> of Three Dimensions<br />
27 Infinitely many<br />
28 One<br />
29 Infinitely many<br />
30 Infinitely many<br />
31 One<br />
32 One<br />
33 One<br />
34 None<br />
35 One<br />
CHAPTER<br />
11<br />
11-2 Perpendicular Lines,<br />
Planes, and Dihedral<br />
Angles<br />
(pages 257–258)<br />
1 False<br />
2 True<br />
3 True<br />
4 False<br />
5 True<br />
6 True<br />
7 False<br />
8 True<br />
9 One<br />
10 Infinitely many<br />
11 Four<br />
12 Infinitely many<br />
13 One<br />
14 One<br />
15 a ABCD b GFE<br />
16 a BCDE and GCDE<br />
b PRS, SRT, ADE, and EDF<br />
17 The plane that bisects <strong>the</strong> dihedral angle<br />
18 −−−<br />
QR , −−<br />
QS , −−−<br />
QU , −−−<br />
WP , −−<br />
TP , −−<br />
VP
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
19 1. BC plane M<br />
2. BCA and BCD are right angles.<br />
3. BCA BCD<br />
4. −−<br />
AB −−<br />
DB<br />
5. −−<br />
BC −−<br />
BC<br />
6. BCA BCD (HL HL)<br />
7. BAC BDC (CPCTC)<br />
11-3 Parallel Lines and<br />
Planes<br />
(pages 259–260)<br />
1 Parallel<br />
2 Perpendicular<br />
3 Perpendicular<br />
4 Parallel<br />
5 Parallel<br />
6 Parallel<br />
7 Parallel<br />
8 Infinitely many<br />
9 None<br />
10 One<br />
11 Line a is parallel to plane M. Line b is contained<br />
in plane R. Line b is in plane M. Lines<br />
a and b are coplanar and do not intersect.<br />
There<strong>for</strong>e, a b.<br />
12 True<br />
13 True<br />
14 True<br />
15 False<br />
16 False<br />
17 True<br />
18 False<br />
19 False<br />
20 True<br />
21 True<br />
22 True<br />
23 True<br />
24 True<br />
25 True<br />
26 False<br />
27 False<br />
28 False<br />
29 False<br />
30 True<br />
11-4 Surface Area of<br />
a Prism<br />
(pages 264–265)<br />
1 (3) 80<br />
2 a 104 b 108 c 224<br />
3 4 faces, 7 edges, 4 vertices<br />
4 5 faces<br />
5 Right prism<br />
6 a 404 cm 2 b 223 ft 2<br />
7 208<br />
8 242 square inches<br />
9 588<br />
10 435 in. 2<br />
11 Lateral area: 255 cm 2 ; total area:<br />
25 √ 3<br />
255 _ 2<br />
c m<br />
2<br />
12 Lateral area: 45 cm 2 ; total area:<br />
9 √ 3<br />
45 _ 2<br />
c m<br />
2<br />
13 144 32 √ 3 cm 2<br />
14 3,880 in. 2<br />
15 Lateral area: 72; total area: 72 12 √ 3<br />
16 Lateral area: 510 square inches; total area:<br />
510 75 √ 3 square inches<br />
17 3 √ 3 in.<br />
18 d 2 h 2 (AB) 2 , but (AB) 2 l 2 w 2 . By substitution,<br />
d 2 h 2 l 2 w 2<br />
11-5 Symmetry Planes<br />
(pages 267–268)<br />
1 (1) A<br />
2 (2) E<br />
3 (1) F<br />
Exercises 4–7: Check students’ sketches.<br />
8 7 symmetry planes<br />
9 Check students’ sketches.<br />
10 112.5 √ 3 <br />
11-5 Symmetry Planes 69
11-6 Volume of a Prism<br />
(pages 269–270)<br />
1 288<br />
2 15,625 m 3<br />
3 a Volume: 105 cm 3 ; surface area: 142 m 2<br />
b Volume: 3,600 in. 3 ; surface area: 1,500 in. 2<br />
c Volume: 48 ft 3 ; surface area: 88 ft 2<br />
4 147<br />
5 a 2 b 6 c 4<br />
d 5 e 3<br />
6 14<br />
7 1 _<br />
2<br />
8 Volume: 1,920 ft 3 ; surface area: 992 ft 2<br />
9 1<br />
10 Volume: 2,197 ft 3 ; diagonal: 13 √ 3 ft<br />
11 30 inches<br />
12 480 cm 3<br />
13 9<br />
14 2.7 ft<br />
15 x √ 3<br />
16 81 √ 3<br />
17 Volume: 343; surface area: 294<br />
18 24 √ 3 in. 3<br />
19 Volume is ten times as large.<br />
20 a Each solid has a volume of 360 √ 3 cubic<br />
units.<br />
b triangular prism 360 square units,<br />
hexagonal prism 120 √ 6<br />
21 Diagonal, d √ <br />
l 2 w 2 h 2 , so d 2 <br />
l 2 w 2 h 2 . Multiply by four, so that<br />
4 d 2 4 l 2 4 w 2 4 h 2 .<br />
11-7 Cylinders<br />
(pages 273–274)<br />
1 (3) 6 inches<br />
2 h 11<br />
3 V _<br />
9<br />
4 r 1 _<br />
4<br />
5 h 1 _<br />
<br />
6 a 112 cm 2 b 136.5 cm 2 c 196 cm 3<br />
7 a 96 in. 2 b 168 in. 2 c 288 in. 2<br />
8 a 24 in. 2 b 26 in. 2 c 12 in. 2<br />
9 a 6 m 2 b 14 m 2 c 6 in. 2<br />
10 a r 3 b r 4<br />
70 Chapter 11: <strong>Geometry</strong> of Three Dimensions<br />
11 A cylinder with a radius of 8 and a height of<br />
12 has a volume of 768. A cylinder with a<br />
radius of 12 and a height of 8 has a volume<br />
of 1,152. Their difference is 384.<br />
12 38 pounds<br />
13 r 3.4<br />
14 h 29.4<br />
15 a 64 b The volume is twice as<br />
large; 128<br />
c 32 d The lateral area stays<br />
<strong>the</strong> same; 32.<br />
16 a Doubled b Volume is 1 _ of <strong>the</strong><br />
4<br />
volume original.<br />
17 Yes. The volume of <strong>the</strong> glass with a diameter<br />
of 2.4 is 1.44. Double <strong>the</strong> volume is 2.88.<br />
The volume of <strong>the</strong> second glass is 2.89,<br />
which is greater than twice <strong>the</strong> volume of <strong>the</strong><br />
first glass.<br />
18 False<br />
19 True<br />
20 Lateral surface area: 960 in. 2 ; volume:<br />
3,840 in. 3<br />
21 h 6,930 _<br />
36<br />
385<br />
_<br />
2<br />
22 16 : 64 or : 4<br />
23 a V 256 b V 512<br />
c V 256 512<br />
11-8 Pyramids<br />
(pages 278–279)<br />
1 234 cm 2<br />
2 240 cm 2<br />
3 1,152 in. 2<br />
4 4,848 cm 3<br />
5 57.2 in. 3<br />
6 7.5 ft 3<br />
7 48 √ 3<br />
8 h 300 ft<br />
9 h 6<br />
10 279 ft 2<br />
11 864 in. 3<br />
12 120<br />
13 a 180 25 √ 3 b 340<br />
c 360 150 √ 3<br />
14 9 √ 3 <br />
15 Volume of pyramid is 80. Volume of prism<br />
is 240.
16 240<br />
17 144 √ 3 cm 2<br />
18 432 √ 3 ft 3<br />
11-9 Cones<br />
(pages 283–284)<br />
1 Lateral area: 60; total area: 96;<br />
volume: 96<br />
2 Lateral area: 20; total area: 36;<br />
volume: 16<br />
3 Lateral area: 136; total area: 200; volume:<br />
320<br />
4 Lateral area: 80; total area: 105<br />
5 Lateral area: 135 cm 2 ; volume: 324 cm 3<br />
6 Lateral area: 260 in. 2 ; total area: 360 in. 2 ;<br />
volume: 800 in. 3<br />
7 r 3<br />
8 V 36<br />
9 V 16<br />
10 V 12<br />
11 r 10<br />
12 1 _<br />
3 base height<br />
13 equal<br />
14 three times<br />
15 one-third<br />
16 No, because <strong>the</strong> slant height is always<br />
greater than <strong>the</strong> radius.<br />
17 4 : 1<br />
18 h 8<br />
19 28.5 in. 3<br />
20 Volume: 3,056 _<br />
3 1,018.67 in. 3 ; lateral area:<br />
300 in. 2<br />
21 144 √ 3<br />
22 60<br />
11-10 Spheres<br />
(pages 287–288)<br />
1 a r 10.5 ft b r 6.2 ft c r 2.9 ft<br />
2 a r 5 mi b r 2 √ 5 4.5 mi<br />
c r 4 mi d r 1.6 mi<br />
3 a 288 in. 3 b 36 ft 3<br />
c 972,000 mi 3 d 1 _ yd<br />
6 3<br />
e 4 _ 3<br />
m<br />
81<br />
4 r 7 in.<br />
5 Surface area: 100 cm 2 ; volume: 500 _ 3<br />
cm<br />
3<br />
6 Surface area: 192 in. 2 ; volume: 256 √ 3 in. 3<br />
7 Volume: 1,679,616,000 cubic miles; surface<br />
area: 4,665,600 square miles<br />
8 63,361,600 square miles<br />
9 1,000,000 times<br />
10 r 5<br />
11 r 3<br />
12 9 : 25 or 9 _<br />
25<br />
13 a 2 : 5 or 2 _<br />
5<br />
14 a 2 _<br />
b<br />
3<br />
4 _<br />
9<br />
15 a m 2 : n 2 b m 3 : n 3<br />
16 64 _<br />
27<br />
17 108 ounces<br />
18 1,458 lb.<br />
b 4 : 25 or 4 _<br />
19 a r √ 2 _<br />
b r 4 _<br />
20 r _ √ 24<br />
<br />
√ <br />
21 a 1 _ ft<br />
6 3 b 4 _ ft<br />
3 3 c 4.5 ft 3<br />
22 64<br />
23 h 4r<br />
24 : 6 or _<br />
6<br />
25 2 : 3 or 2 _<br />
3<br />
26 They are equal.<br />
Lateral area of cylinder 16 in. 2<br />
Surface area of sphere 16 in. 2<br />
25<br />
Chapter Review (pages 289–290)<br />
1 (4) 216 in. 2<br />
2 False<br />
3 False<br />
4 True<br />
5 True<br />
6 False<br />
7 False<br />
8 False<br />
9 False<br />
10 True<br />
11 True<br />
12 a 8 vertices, 6 faces, 12 edges<br />
b 6 vertices, 5 faces, 9 edges<br />
c 8 vertices, 6 faces, 12 edges<br />
d 10 vertices, 7 faces, 15 edges<br />
13 a Surface area: 216, volume: 432<br />
b Surface area: 90, volume: 100<br />
c Surface area: 52 ft 2 , volume: 24 ft 3<br />
Chapter Review 71
14 47.5 in. 2<br />
15 h 4 in.<br />
16 25 cm 3<br />
17 The cylinder <strong>for</strong>med by spinning <strong>the</strong> rectangle<br />
about <strong>the</strong> shorter side is 96 cm 3 greater<br />
in volume.<br />
r 6 and h 8, V 288 cm 3<br />
r 8 and h 6, V 384 cm 3<br />
18 36 9 √ 3 in. 2<br />
19 36 √ 3 in. 3<br />
20 144 √ 3 cm 3<br />
21 480<br />
22 156 in. 2<br />
23 √ 65 in.<br />
Ratios, Proportion,<br />
and Similarity<br />
12-1 Ratio and Proportion<br />
(pages 294–295)<br />
1 No<br />
2 Yes<br />
3 No<br />
4 Yes<br />
5 a 3 _<br />
2<br />
b 2 _<br />
5<br />
c 3 _<br />
5<br />
d 5 _<br />
3<br />
6 77<br />
7 15<br />
8 9<br />
9 15<br />
10 5<br />
72 Chapter 12: Ratios, Proportion, and Similiarity<br />
24 100 ft 3<br />
25 60 ft 2<br />
26 Lateral area: 15 in. 2 ; volume: 12 in. 3<br />
27 r 3 in.; volume: 36 in. 3<br />
28 Surface area: 40,000 cm 2<br />
Volume: 4,000,000 _<br />
3 cm 3 or 1,333,333 1 _ cm<br />
3 3<br />
29 Surface area: 4 ft 2 ; volume: 4 _ ft<br />
3 3<br />
30 r 8 _<br />
√ meters<br />
31 5 _ inches or 2.5 inches<br />
2<br />
32 40.57 41 scoops<br />
11 44<br />
12 15<br />
13 39<br />
14 mt _<br />
a<br />
15 2ma _<br />
r<br />
16 4ar _<br />
t<br />
17 9<br />
18 8<br />
19 15<br />
20 4 √ 3 <br />
21 3 √ 11<br />
22 5 √ 3 <br />
23 6 √ 2 <br />
24 4 √ 6 <br />
25 1 _<br />
10<br />
26 1 _ or _ √ 3<br />
3 √ 3 9<br />
CHAPTER<br />
12
27 40, 50<br />
28 70, 110<br />
29 20, 35<br />
30 16, 20, 24<br />
31 length 28, width 49<br />
32 45, 135<br />
33 a 8 b 6 c 10<br />
34 a 20 b 9<br />
12-2 Proportions Involving<br />
Line Segments<br />
(pages 298–300)<br />
1 a–d are all true<br />
2 2 _ <br />
5 4 _ Proportions are not true, lines not<br />
8<br />
parallel.<br />
3 6<br />
4 28<br />
5 5.4<br />
6 6<br />
7 24<br />
8 5<br />
9 ac _<br />
b<br />
10 np<br />
_<br />
m<br />
11 6<br />
12 20<br />
c(a b)<br />
13 _<br />
a<br />
14 No<br />
15 No<br />
16 No<br />
17 No<br />
18 Yes<br />
19 12<br />
20 1 : 3 or 1 _<br />
3<br />
21 11 cm<br />
22 x 6, AB 47, DE 17, AC 34, EC 9,<br />
BE 9, AD 23.5, DB 23.5<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
23 a 1. ABC<br />
2. P is <strong>the</strong> midpoint of −−<br />
AB .<br />
3. Q is <strong>the</strong> midpoint of −−<br />
BC .<br />
4. −−<br />
PQ −−<br />
AC (A line joining <strong>the</strong> midpoints<br />
of two sides of a triangle is<br />
parallel to <strong>the</strong> third side.)<br />
5. −−<br />
PQ ___<br />
AR<br />
6. R is <strong>the</strong> midpoint of −−<br />
AC .<br />
7. −−<br />
AB −−−<br />
RQ<br />
8. −−<br />
AP −−−<br />
RQ<br />
9. PQRA is a parallelogram. (Definition<br />
of a parallelogram)<br />
b 30<br />
24 1. ABC, ADC<br />
2. −− wx −−<br />
AC (A line dividing two<br />
sides of a triangle proportionally<br />
is parallel to<br />
<strong>the</strong> third side.)<br />
3. −− zy −−<br />
AC<br />
4. −− wx −− zy<br />
5. BAD, BCD<br />
6. −− wz −−<br />
BD<br />
7. −− xy −−<br />
BD<br />
8. −− wz −− xy<br />
9. wxyz is a ( Definition of a<br />
parallelogram. parallelogram)<br />
25 48 √ 3 <br />
12-3 Similar Polygons<br />
(pages 302–303)<br />
1 1 : 1<br />
2 4, 4.5, 5<br />
3 1 : 4<br />
4 45, 55<br />
5 a 2 : 3<br />
b 4.5, 7.5<br />
c 18<br />
6 9, 13.5, 18<br />
7 5a, 5b, 5c<br />
8 All corresponding angles are congruent, all<br />
corresponding sides are in proportion, 1 : 2.<br />
9 w _ ,<br />
3 x _ ,<br />
3 y<br />
_ ,<br />
3 z _<br />
3<br />
10 32, 40<br />
11 mR mQ 110<br />
12 mI 40; mK 160<br />
13 12<br />
12-3 Similar Polygons 73
14 a 3 : 2 b QR 10, RS 20,<br />
ST 12, PT 22<br />
15 a True b False c True<br />
d True e False f True<br />
g False h False<br />
16 If <strong>the</strong> vertex angles are congruent, <strong>the</strong>n <strong>the</strong><br />
base angles must also be congruent. Similarly,<br />
if <strong>the</strong> base angles are congruent, <strong>the</strong>n<br />
<strong>the</strong> vertex angles are congruent. Triangles are<br />
similar by (AA) or (AAA).<br />
12-4 Proving Triangles<br />
Similar<br />
(pages 307–308)<br />
1 (2) similar<br />
2 Vertical angles are congruent. (AA)<br />
3 Not similar<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
4 a 1. −−<br />
AE −−<br />
BC<br />
2. BCD DAE ( Alternate interior<br />
angles)<br />
3. −−<br />
AC and −−<br />
BE intersect at D.<br />
4. BDC ADE (Vertical angles)<br />
5. ADE CDB (AA)<br />
b AD 30<br />
5 All corresponding sides are in<br />
proportion; AB _ <br />
BE<br />
CD _ <br />
AE<br />
CE _ <br />
4<br />
DE _ . Triangles<br />
5<br />
are similar.<br />
6 4 _ and<br />
6 5 _ , corresponding sides are not in<br />
3<br />
proportion. Triangles are not similar.<br />
7 6 _ and<br />
7 4 _ , corresponding sides are not in<br />
5<br />
proportion. Triangles are not similar.<br />
8 1. −−−<br />
CD −−<br />
AB<br />
2. CDA and CDB are right angles.<br />
3. CDA CDB<br />
4. CAD CBD<br />
5. CAD CBD (AA)<br />
74 Chapter 12: Ratios, Proportion, and Similiarity<br />
9 1. −−<br />
AB −−<br />
DE<br />
2. EDF BAC (Corresponding<br />
angles are<br />
congruent.)<br />
3. −−<br />
BC −−<br />
EF<br />
4. EFD BCA<br />
5. ABC DEF (AA)<br />
10 1. −−<br />
BE −−<br />
AC and −−−<br />
CD −−<br />
AB<br />
2. HEC and HDB are right angles.<br />
3. HEC HDB<br />
4. DHB CHE (Vertical angles are<br />
congruent.)<br />
5. EHC DHB (AA)<br />
11 1. −−<br />
DE −−<br />
BC and −−<br />
DF −−<br />
AB<br />
2. DEC and DFA are right angles.<br />
3. DEC DFA<br />
4. Parallelogram ABCD<br />
5. FAD ECD<br />
6. AFD CED (AA)<br />
12 1. DBE ADF<br />
2. BAC BDE (Corresponding<br />
angles)<br />
3. B B<br />
4. DBE ABC (AA)<br />
13 1. Parallelogram ABCE<br />
2. −−−<br />
AED<br />
3. −−<br />
BC −−<br />
AE<br />
4. −−<br />
BC −−−<br />
AD<br />
5. CBF ADB (Alternate interior<br />
angles)<br />
6. BAD FCB<br />
7. BAD FCB (AA)<br />
14 1. ABC PQR<br />
2. ABC PQR<br />
3. −−<br />
BD bisects ABC.<br />
4. −−<br />
QS bisects PQR.<br />
5. ABD PQS<br />
6. BAD QPS<br />
7. ABD PQS (AA)<br />
15 1. Parallelogram ABCD<br />
2. −−−<br />
AD −−<br />
BC<br />
3. GAE BCG (Alternate interior<br />
angles are<br />
congruent.)<br />
4. −−<br />
AC bisects HAE.<br />
5. HAG GAE<br />
6. HAG BCG<br />
7. −−−<br />
AH −−<br />
AE
8. GEA GHA<br />
9. GEA GBC<br />
10. GHA GBC<br />
11. HAG BCG<br />
12-5 Dilations<br />
(pages 310–311)<br />
1 a 10 _<br />
80<br />
b<br />
3<br />
_ c 9<br />
3<br />
2 a BC 10, PA 2.2, BA 11, PS 9.6<br />
b 10 _ 5<br />
<br />
8 _<br />
4<br />
3 (21, 9)<br />
4 (6, 12)<br />
5 (9, 0)<br />
6 (27, 3)<br />
7 (12, 12)<br />
8 (10, 36)<br />
9 (20, 12)<br />
10 (2.5, 1)<br />
11 (4, 2)<br />
12 (3, 0)<br />
13 (3, 3.5)<br />
14 (2 √ 2 , 2.5)<br />
15 (16, 8)<br />
16 (10, 2)<br />
17 (15, 0)<br />
18 (3, 15)<br />
19 (3, 2)<br />
20 (4, 8)<br />
21 D 2 r x-axis<br />
22 D 3 r y-axis<br />
23 r x-axis D 1 _<br />
2<br />
24 r x-axis D 1 _<br />
4<br />
25 a A(0, 0), B(12, 0), C(15, 6), D(3, 6)<br />
b Slope AB 0; slope BC 2; slope CD 0;<br />
slope DA 2<br />
c Midpoint M (2.5, 1) and midpoint<br />
M (7.5, 3). Yes, M is <strong>the</strong> image result<br />
of D 3 operating on point M. Midpoints are<br />
preserved under dilation.<br />
12-6 Proving Proportional<br />
Relationships Among<br />
Segments Related to<br />
Triangles<br />
(pages 315–316)<br />
1 3 : 4 and 3 : 4<br />
2 5 cm<br />
3 4 : 9<br />
4 11 in.<br />
5 AD 6, DC 8<br />
6 18<br />
7 15<br />
8 4 : 7<br />
9 AC 23, PR 8<br />
10 RQ 6, BC 12.2, AC 20.6<br />
11 13<br />
12 19 _ or 4.75<br />
4<br />
13 22<br />
14 12<br />
15 a 4 : 7 b 16 and 20<br />
c 84 and 48 d 48 _ <br />
4<br />
84 _<br />
7<br />
16 a BQ 8, QR 5, PR 8 b 36<br />
12-7 Using Similar Triangles<br />
to Prove Proportions<br />
or to Prove a Product<br />
(pages 319–321)<br />
1 128<br />
2 40<br />
3 108<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
4 a 1. −−<br />
ED −−<br />
AC and −−<br />
AB −−<br />
CB<br />
2. EDA and ABC are right angles.<br />
3. EDA ABC<br />
12-7 Using Similar Triangles to Prove Proportions or to Prove a Product 75
4. A A<br />
5. ADE ABC<br />
b<br />
(AA)<br />
AC _ <br />
AB<br />
AE _<br />
AD<br />
c 32<br />
5 a Each smaller triangle is similar to <strong>the</strong><br />
larger triangle so <strong>the</strong>y are similar to each<br />
o<strong>the</strong>r.<br />
b Corresponding sides of similar triangles<br />
are in proportion.<br />
c Product of means product of extremes.<br />
d BD 6<br />
6 1. MAH is a right angle.<br />
2. −−<br />
UT −−−<br />
AH<br />
3. UTH is a right angle.<br />
4. MAH UTH<br />
5. H H<br />
6. MAH UTH (AA)<br />
7. MA _ <br />
AH<br />
UT _<br />
TH (Corresponding<br />
sides of similar<br />
triangles are in<br />
7 1.<br />
proportion.)<br />
−−<br />
AB −−<br />
DE<br />
2. CAB CED (Alternate interior<br />
angles are<br />
congruent.)<br />
3. ACB ECD (Vertical angles are<br />
congruent.)<br />
4. ACB ECD (AA)<br />
5. AB _ <br />
ED<br />
AC _<br />
EC<br />
6. AB _ AC<br />
<br />
ED _<br />
EC (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
8 1. EBA CDA<br />
2. A A<br />
3. EBA CDA<br />
4. BCF DEF<br />
5. −−<br />
AC −−<br />
AE<br />
6. DEC BCE (Isosceles triangle<br />
<strong>the</strong>orem)<br />
7. DEC DEF (Subtraction<br />
BCE BCF<br />
8. FEC FCE<br />
9. CBD EDB<br />
postulate)<br />
10. CBD EDB<br />
11.<br />
(AA)<br />
BC _ <br />
BE<br />
DE _<br />
DC<br />
76 Chapter 12: Ratios, Proportion, and Similiarity<br />
12. BC DC (The product of <strong>the</strong><br />
BE DE means equals <strong>the</strong><br />
product of <strong>the</strong><br />
extremes.)<br />
9 1. EBC CDE<br />
2. BFC DFE (Vertical angles are<br />
congruent.)<br />
3. DEF BCF (AA)<br />
4. FB _ <br />
FD<br />
FC _<br />
FE<br />
5. FB _ FC<br />
<br />
FD _ (Corresponding sides<br />
FE<br />
of similar triangles are<br />
in proportion.)<br />
10 1. −−<br />
AE −−<br />
BC<br />
2. ADC and EDC are right angles.<br />
3. ADC EDD<br />
4. DAC DEC<br />
5. DEC DAC (AA)<br />
6. EC _ AC<br />
_<br />
ED<br />
AD<br />
7. EC AD (The product of <strong>the</strong><br />
AC ED means equals <strong>the</strong><br />
product of <strong>the</strong><br />
extremes.)<br />
11 1. Isosceles triangle WXZ with −−−<br />
WX −−−<br />
WZ<br />
2. X Z<br />
3. YTW YRW<br />
4. XTY ZRY (Supplements of congruent<br />
angles are<br />
congruent.)<br />
5. XTY ZRY (AA)<br />
6. YT _ <br />
YR<br />
XY _<br />
ZY<br />
7. YT _ <br />
XY<br />
YR _ (Corresponding sides<br />
ZY<br />
of similar triangles are<br />
in proportion.)<br />
12 1. Rectangle DEFG<br />
2. DGB and EFC are right angles.<br />
3. DGB EFC<br />
4. Isosceles triangle ADE<br />
5. BDG CEF (Supplements of congruent<br />
angles are<br />
congruent.)<br />
6. BDG CEF (AA)<br />
7. DG _ BG<br />
<br />
EF _ (Corresponding sides<br />
CF<br />
of similar triangles are<br />
in proportion.)
12-8 Proportions in a<br />
Right Triangle<br />
(pages 323–324)<br />
1 (1) 10<br />
2 (3) √ 21<br />
3 (3) 4<br />
4 (1) √ 77<br />
5 a False b True c False<br />
d False e True f True<br />
g True<br />
6 2 √ 15<br />
7 5<br />
8 6<br />
9 √ 55<br />
h False<br />
10 x 2 √ 5 , y 4 √ 5 , z 4<br />
11 x 2 √ 5 , y 4 √ 5 , z 8<br />
12 x 3 √ 6 , y 6 √ 3 , z 6<br />
13 x 4 √ 3 , y 8 √ 3 , z 12<br />
14 x 20, y 8 √ 6 , z 40 √ 6<br />
15 x 9, y 6.75, z 18.75<br />
16 x 4 √ 5 , y 10<br />
17 x 3 √ 10 , y 3 √ 6 , z 3 √ 15<br />
18 x 12, y 4 √ 5 , z 6 √ 5<br />
19 x 9, y 6, z 6 √ 3<br />
20 x 25, y 17.5, z 29.2<br />
21 x 2.4, y 3.2, z 1.8<br />
22 a x 21 b (10) 2 23 13<br />
x(x 21) c 4<br />
24 a LR 2 √ 26 1, LM 4 √ 26<br />
b 9 and 10<br />
12-9 Pythagorean Theorem<br />
and Special Triangles<br />
(pages 327–329)<br />
1 (2) 10 √ 2 <br />
2 (3) 25 feet<br />
3 (1) √ 13<br />
4 (1) 50 miles<br />
5 (4) √ 41<br />
6 (1) 24 feet<br />
7 (2) 13<br />
8 (4) {10, 24, 26}<br />
9 14<br />
10 b, c, d, e, f, g<br />
11 a 3 √ 2 b √ 3 c<br />
d 9 e 5 _<br />
12<br />
f<br />
√ <br />
5<br />
√ <br />
14<br />
12 6<br />
13 8<br />
14 240 cm 2<br />
15 32 in.<br />
16 x 10 √ 5 , 2x 20 √ 5<br />
17 24<br />
18 5<br />
19 19.2 in.<br />
20 Perimeter is 56. Area is 192.<br />
21 3<br />
22 10<br />
23 3 √ 3 <br />
24 BD 2 a 2 b 2<br />
BD 2 d 2 c 2<br />
a 2 b 2 d 2 c 2<br />
a 2 d 2 b 2 c 2<br />
25 AB 2 BC 2 AC 2<br />
AC 2 AD 2 CD 2<br />
AB 2 BC 2 AD 2 CD 2<br />
Special Triangles<br />
(pages 332–334)<br />
1 (1) 6 √ 2 ft and 6 √ 2 ft<br />
2 (3) 1 : √ 3 : 2<br />
3 (4) 6 √ 2 <br />
4 (1) 1<br />
5 (2) 6 √ 3 <br />
6 (1) 30<br />
7 (3) 3.75<br />
8 (2) 2 √ 6 <br />
9 (3) 60<br />
10 a BC 7, AC 7 √ 3 <br />
b BC 2.5, AC 2.5 √ 3<br />
c BC 3 √ 3 , AC 9<br />
d AC 9 √ 3 , AB 18<br />
e AC 12, AB 8 √ 3<br />
f BC 7, AB 14<br />
g BC 2.75 √ 3 , AC 8.25<br />
h BC 4 √ 3 , AB 8 √ 3 <br />
11 a AC 4, AB 4 √ 2<br />
b BC 10, AB 10 √ 2<br />
c AC 8, BC 8<br />
d AC 1.5, BC 1.5<br />
e AC 6 √ 2 , BC 6 √ 2 <br />
f BC 3 √ 2 , AB 6<br />
g BC 10 √ 2 , AB 20<br />
h AC 7.5 √ 2 , BC 7.5 √ 2 <br />
i BC 4 √ 3 , AB 4 √ 6 <br />
j AC 4 √ 6 , BC 4 √ 6 <br />
12-9 Pythagorean Theorem and Special Triangles 77
12 7 √ 2<br />
13 4<br />
14 5 √ 3<br />
15 x 6 √ 3 , y 12, z 6 √ 2<br />
16 a 6 by 6 √ 3<br />
b 12 12 √ 3 <br />
c 36 √ 3<br />
17 9 √ 2<br />
18 46<br />
19 AB 16 √ 3 , AC 24, DC 8, DB 16<br />
20 2.28<br />
21 a 8<br />
b 2<br />
12-10 Perimeters, Areas,<br />
and Volumes of Similar<br />
Figures (Polygons and<br />
Solids)<br />
(pages 336–337)<br />
1 (1)1 : 3<br />
2 (3) 3 : 5<br />
3 (1) 28<br />
4 a 9 : 1 b 16 : 81 c 81 : 4<br />
d 25 : 1 e 2 : 3<br />
5 a 2 : 3 b 6 : 5 c 1 : 7<br />
d 2 : 5<br />
6 1 : 9<br />
7 343<br />
8 13.5<br />
9 25 : 1<br />
10 539 ft<br />
e 9 : 11<br />
2<br />
11 1 : 25<br />
12 27 : 64<br />
13 48<br />
14 27 ft 3<br />
15 27.7 in.<br />
16 506.25 kg<br />
17 a 1 : 9 b 1 : 3<br />
18 a Yes b 2 : 3 c 4 : 9 d 8 : 27<br />
e Ratio of surface areas: 312 _ <br />
4<br />
702 _<br />
9<br />
Ratio of volumes: 360 _ 8<br />
<br />
1,215 _<br />
27<br />
19 72<br />
20 a 2.4 b 27 : 125<br />
78 Chapter 12: Ratios, Proportion, and Similiarity<br />
Chapter Review (pages 338–341)<br />
1 (1) 4 : 25<br />
2 (4) 5 : 14<br />
3 (3) 15<br />
4 (2) x a _<br />
a<br />
5 (3) √ 3 <br />
6 (4) 64 pounds<br />
7 a 6 b 6 c an _<br />
8 a 5 : 6 b 2 : 9<br />
m<br />
c n : m d c : (a b)<br />
e p : (m n) f (m n) : (a b)<br />
g (n b) : (a m)<br />
9 a 6 b 8a c 10 _<br />
10 30<br />
11 11.25<br />
12 6 : 35<br />
13 6<br />
14 6<br />
15 14<br />
3<br />
16<br />
ac bc _<br />
a<br />
17<br />
8 _ 5<br />
<br />
8 x _ 8<br />
, EC <br />
x 4 _ , AC <br />
3 20 _<br />
3<br />
18 a 3<br />
19 54<br />
b Expansion<br />
20 102<br />
21 Yes, <strong>the</strong> ratio of similitude is 3 _ .<br />
2<br />
22 Yes, <strong>the</strong> ratio of similitude is 1 _<br />
23 mQ 125, PQ 6 _ , QR 2<br />
5<br />
24 2<br />
25 13<br />
26 BD 12, AB 4 √ 13 , BC 6 √ 13<br />
27 20, 21, 29<br />
28 a 5 √ 2 <br />
b 50<br />
29 34<br />
30 x 40 √ 3 , y 80<br />
31 x 8 √ 3 , y 8<br />
32 x 6 √ 3 , y 9<br />
33 √ 2<br />
34 126<br />
35 75 ft 2<br />
36 8 _<br />
27<br />
37 80 in. 3<br />
38 81<br />
39 45<br />
40 18<br />
41 72 √ 3 in. 2<br />
42 Area 60, perimeter 24 10 √ 2 <br />
2 .
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
43 1. −−<br />
TS −−<br />
QP<br />
2. QRP TSR (Corresponding<br />
angles are<br />
congruent.)<br />
3. PQR STR<br />
4. R R<br />
5. PQR STR (AAA AAA)<br />
44 1. −−<br />
AE −−<br />
BC , −−<br />
BD −−<br />
AC<br />
2. FEB and BDC are right angles.<br />
3. FEB BDC<br />
4. FBE FBE<br />
5. BFE BCD (AA)<br />
45 1. −−<br />
AB −−−<br />
CD<br />
2. ABE DCE (Alternate interior<br />
angles are<br />
congruent.)<br />
<strong>Geometry</strong> of <strong>the</strong> Circle<br />
13-1 Arcs and Angles<br />
(pages 345–346)<br />
1 a 36 b 51 c 90<br />
d 180 e 2r<br />
2 a BC , CD , AD , BD , AB<br />
b ABC , ADC<br />
c ABD , BDA , ADB<br />
3. BAE CDE<br />
4. AEB CED (Vertical angles are<br />
congruent.)<br />
5. a ABE DCE (AAA AAA)<br />
6. BE _ EC<br />
<br />
AE _<br />
ED<br />
7. b BE _ <br />
AE<br />
EC _<br />
ED (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
8. c BE ED (The product of<br />
EC AE means equals<br />
<strong>the</strong> product of<br />
46 1.<br />
extremes.)<br />
−−<br />
BC −−<br />
AF<br />
2. ECA and FDA are right angles.<br />
3. ECA FDA<br />
4. A A<br />
5. ADF ACB<br />
6. BA _ =<br />
AF _<br />
BC<br />
DF<br />
7. BA DF (The product of<br />
AF DC means equals<br />
<strong>the</strong> product of<br />
extremes.)<br />
CHAPTER<br />
13<br />
3 RB 105, BS 75, AS 105, AR 75<br />
4 a 155 b 25 c 155 d 335<br />
5 a 296 b 244 c 296 d 244<br />
6 a 73 b 132 c 155 d 180<br />
e 155<br />
i 287<br />
f 228 g 253 h 205<br />
7 a 20 b 60 c 120 d 110<br />
e 7 f 60 g 70 h 120<br />
i 110 j 230 k 240 l 290<br />
13-1 Arcs and Angles 79
13-2 Arcs and Chords<br />
(pages 350–351)<br />
1 a 16 b 7.5 c m _<br />
d 3 √ 6 e 5 √ 2<br />
2<br />
2 a True<br />
3 109<br />
b True c False<br />
4 a 90 b 72 c 60 d 36<br />
5 a 24 b 22.5 c 20<br />
d 18 e 15<br />
6 a 10 b 5 √ 2 c 5<br />
7 a 6<br />
8 6 in.<br />
9 17<br />
b 6 √ 2 c 12 d 6 √ 2 <br />
10 a 15<br />
11 6.<br />
12 8 √ 2<br />
13 x √ 2<br />
14 5<br />
15 25<br />
b 7.5<br />
16 a 30 b 60 c 60<br />
d 60 e 60 f 300 g 150<br />
17 a 30 b 60 c 10<br />
d 5 e 5 √ 3<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
18 1. Radii −−−<br />
OK and −−−<br />
OM<br />
2. AOL DOL (Central angles are<br />
congruent.)<br />
3. −−<br />
LA −−−<br />
OK , −−<br />
LD −−−<br />
OM<br />
4. LAO and LDO are right angles.<br />
5. LAO and LDO are right triangles.<br />
6. LAO LDO<br />
7. −−<br />
LO −−<br />
LO<br />
8. LAO LDO (Leg-angle)<br />
19 1. ABD CBD<br />
2. −−−<br />
OA −−<br />
OB<br />
3. BAO ABD<br />
4.<br />
(All radii of <strong>the</strong><br />
same circle are<br />
congruent.)<br />
−−−<br />
OC −−<br />
OB<br />
5. BCO CBD<br />
80 Chapter 13: <strong>Geometry</strong> of <strong>the</strong> Circle<br />
6. AOB COB<br />
7. AOD COD<br />
8. AD CD (Congruent inscribed<br />
angles intercept congruent<br />
arcs.)<br />
20 1. −−−<br />
OA , −−<br />
OB , and −−−<br />
OC are radii of circle O.<br />
2. BAC BCA<br />
3. AB BC (Congruent inscribed<br />
angles intercept congruent<br />
arcs.)<br />
4. AOB COB (Central angles are<br />
equal to <strong>the</strong>ir arcs.)<br />
13-3 Inscribed Angles<br />
and Their Measure<br />
(pages 354–357)<br />
1 a 25 b 55 c 125<br />
d 60.4 e 4<br />
2 a 52 b 124 c 15<br />
d 180 e (16x)<br />
3 mx 70, my 35; 70 35 105<br />
4 a 50 b 130 c 56<br />
d 25<br />
5 53<br />
6 70<br />
7 80<br />
8 35<br />
9 140<br />
e 62 f 25<br />
10 mC 75, mM 82<br />
11 85.5<br />
12 m 1 120, m2 45, m3 75, m4 60<br />
13 m1 57, m2 33, m3 90, m 4 114<br />
14 m 1 74, m 2 74, m3 37, m4 106<br />
15 a 140 b 80 c 40<br />
d 70 e 70<br />
16 m1 57, m2 48, m3 57, m4 48<br />
17 a 80 b 188 c 46 d 94<br />
18 a 64 b 108 c 140 d 70<br />
e 24 f 54 g 94<br />
19 a 86 b 102 c 88<br />
d 56 e 116<br />
20 a 44 b 88 c 44 d 44<br />
e 92 f 88<br />
21 a 60 b 60 c 60 d 30<br />
e 60 f 120 g 30 h 60<br />
i 120 j 60 k 90 l 30
22 Label arcs with degree measures that sum<br />
to 360, as with x, 2x, 3x, 4x. Then find <strong>the</strong><br />
values of <strong>the</strong> inscribed angles.<br />
23 Parallel lines cut congruent arcs x and y.<br />
There<strong>for</strong>e,<br />
2x 2y 360<br />
x y 180<br />
Each inscribed angle is equal to one-half <strong>the</strong><br />
intercepted arc: 90.<br />
13-4 Tangents and Secants<br />
(pages 363–365)<br />
1 a<br />
b<br />
c<br />
d<br />
2 a 12 b 3<br />
3 a 3.25 b 1<br />
4 a disjoint b externally tangent<br />
c intersecting twice<br />
d internally tangent<br />
e disjoint internally<br />
f concentric<br />
5 a Sketch of two circles externally disjoint<br />
b Sketch of two intersecting circles, not<br />
tangent<br />
c Sketch of externally tangent circles<br />
d Sketch of internally tangent circles<br />
6 a 1 b 3<br />
c 2 d 4<br />
7 a 160 b 130 c 90<br />
d 40 e 180 x<br />
8 a 60 b 45 c 35<br />
d 51 e 67.5<br />
180 n<br />
9 a _ b 90 2n c<br />
n<br />
2 _<br />
2<br />
d 45 n e __<br />
180 m n<br />
2<br />
10 16<br />
11 80<br />
12 48<br />
13 234<br />
14 a 18 b 72 c 29.25<br />
15 a 11, 17, 18 b no sides equal<br />
16 a 12, 16, 20<br />
b Satisfies Pythagorean <strong>the</strong>orem:<br />
12 2 16 2 20 2<br />
17 a 10, 15, 15 b two sides are equal<br />
18 BE 4, EC 5, CF 5, AF 6, AC 11,<br />
AB 10<br />
19 AB 20, CB 20, RB 10<br />
20 AB 24, DR 14, DB 32<br />
21 OB 26, DB 36, RB 16<br />
22 OB 20, AB 2 √ 91 , CB 2 √ 91<br />
23 OB 22, CB 8 √ 6 , AB 8 √ 6 <br />
24 AB 20, CB 20, OC 15, OB 25<br />
25 Tangent segments from <strong>the</strong> same point are<br />
congruent. Base angles are equal, each measuring<br />
60. There<strong>for</strong>e, APB is equilateral.<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
26 1. Let E be <strong>the</strong> point of intersection of −−<br />
AB<br />
and −−−<br />
CD .<br />
2. −−<br />
EB and −−<br />
ED are tangent segments to<br />
circle P.<br />
3. −−<br />
EB −−<br />
ED (Two tangent segments<br />
drawn from an external<br />
point are congruent.)<br />
4. −−<br />
EA and −−<br />
EC are tangent segments to<br />
circle O.<br />
5. −−<br />
EA −−<br />
EC<br />
6. −−<br />
EA −−<br />
EB ( Addition postulate)<br />
−−<br />
EC −−−<br />
CD<br />
or −−<br />
AB −−−<br />
CD<br />
13-4 Tangents and Secants 81
27 1. Common external tangents, −−−<br />
AG and −−−<br />
CM<br />
2. Radii −−−<br />
OA and −−−<br />
OC<br />
3. OAG and OCM are right angles.<br />
4. OAG and OCM are right triangles.<br />
5. −−−<br />
OA −−−<br />
OC<br />
6. Tangents −−−<br />
OM and −−−<br />
OG of circle P<br />
7. −−−<br />
OM −−−<br />
OG (Two tangent segments<br />
drawn from an<br />
external point are<br />
congruent.)<br />
8. OAG OCM (HL HL)<br />
9. −−−<br />
AG −−−<br />
CM (CPCTC)<br />
28 1. Circles A and B are congruent with<br />
tangent FG .<br />
2. −−<br />
AF is <strong>the</strong> radius of A. −−<br />
BG is <strong>the</strong> radius of<br />
B.<br />
3. −−<br />
AF −−<br />
BG<br />
4. AFC and BGC are right angles.<br />
5. FCA GCB<br />
6. FCA GCB (AAS AAS)<br />
7. −−<br />
AC −−<br />
BC<br />
29 mA 1 _ (<br />
2 CD BE ). 2(mA) <br />
mCOD mBOE. But mBOE mA.<br />
2(mA) mCOD mA. There<strong>for</strong>e,<br />
3(mA) mCOD.<br />
30 1. −−<br />
AB is tangent to O at E; −−−<br />
CD is tangent to<br />
O at F.<br />
2. −−<br />
OE OF<br />
3. −−−<br />
OA −−−<br />
OC ; −−<br />
OB −−−<br />
OD<br />
4. −−<br />
OE −−<br />
AB<br />
5. OEA and OEB are right angles.<br />
6. −−<br />
OF −−−<br />
CD<br />
7. OFC and OFD are right angles.<br />
8. OEA OFC (HL HL)<br />
OEB OFD<br />
9. −−<br />
AE −−<br />
EB (Addition<br />
−−<br />
CF −−<br />
FD postulate)<br />
or −−<br />
AB −−−<br />
CD<br />
13-5 Angles Formed by<br />
Tangents, Chords, and<br />
Secants<br />
(pages 368–371)<br />
1 (2) 96<br />
2 (1) 24<br />
3 (2) 144<br />
82 Chapter 13: <strong>Geometry</strong> of <strong>the</strong> Circle<br />
4 (4) 173<br />
5 (1) 38<br />
6 (1) 90<br />
7 a 40 b 62 c 225 d 45<br />
e 75 f 100 g 80 h 30<br />
8 a 79 b 38<br />
c x 50, y 60, z 120<br />
9 a 66 b 80 c 73<br />
d 130 e 33 f 73<br />
10 a 90 b 120 c 90 d 60<br />
e 30 f 60 g 30 h 90<br />
11 a 120 b 40 c 80<br />
d 40 e 140 f 100<br />
12 BC CD because in a regular hexagon,<br />
congruent chords subtend congruent arcs.<br />
Chords and a tangent that intercept congruent<br />
arcs are parallel. There<strong>for</strong>e, PC −−<br />
BD<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
13 1. −−<br />
FB −−<br />
EC<br />
2. BEC EBF<br />
3. EHJ BHG<br />
4. EHJ BHG<br />
(Alternate interior angles<br />
are congruent.)<br />
5. BH _ <br />
HE<br />
HG _<br />
HJ<br />
(Corresponding sides<br />
of similar triangles are<br />
in proportion.)<br />
6. BH JH (The product of means<br />
HE HG equals <strong>the</strong> product of<br />
14 1.<br />
extremes.)<br />
−−<br />
BA −−−<br />
DA<br />
2. Diameter −−−<br />
DC<br />
3. BAD and DBC are right angles.<br />
4. −−<br />
AB is tangent to O at B.<br />
5. mDBA 1 _ m<br />
2 BD<br />
6. mDCB 1 _ m<br />
2 BD<br />
7. mDBA mDCB<br />
8. DBA DCB<br />
9. DBA DCB
10. BD _<br />
DA<br />
11. BD _<br />
DC<br />
DC<br />
_<br />
DB<br />
<br />
DA _<br />
BD<br />
( Corresponding sides of<br />
similar triangles are in<br />
proportion.)<br />
12. (BD) 2 ( The product of means<br />
DC DA equals <strong>the</strong> product of<br />
extremes.)<br />
13-6 Measures of Tangent<br />
Segments, Chords and<br />
Secant Segments<br />
(pages 374–376)<br />
1 (4) 20<br />
2 (2) 46<br />
3 (1) 17<br />
4 (3) 24<br />
5 (2) 10<br />
6 (1) 3<br />
7 (3) 25<br />
8 (3) cz _<br />
a<br />
9 (3) 17<br />
10 (3) 33<br />
11 (1) 9<br />
12 24<br />
13 19<br />
14 40<br />
15 44<br />
16 12<br />
17 16<br />
18 NQ 6, QP 16<br />
19 x 4, GH 16<br />
20 −−−<br />
AOB −−−<br />
WZ . Radius 12.5; 10 2 (7.5) 2 <br />
(12.5) 2<br />
13-7 Circle Proofs<br />
(pages 381–382)<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
1 1. A is <strong>the</strong> midpoint of CD ;<br />
−−−<br />
AOB is a diameter.<br />
2. CA AD<br />
3. 1 _ m<br />
2 CA 1 _ m<br />
2 AD<br />
4. mABC 1 _ m<br />
2 AD<br />
mABC 1 _ m<br />
2 CA<br />
5. mABC mABD<br />
6. ACB and ADB are right angles.<br />
7. mACB mADB<br />
8. −−<br />
AB −−<br />
AB<br />
9. ACB ADB (AAS AAS)<br />
10. −−<br />
CB −−<br />
DB (CPCTC)<br />
2 1. −−−<br />
AD −−<br />
CB<br />
2. AD CB<br />
3. mABD 1 _ m<br />
2 AD<br />
mCDB 1 _ m<br />
2 CB<br />
4. mABD mCDB<br />
5. −−<br />
DB −−<br />
DB<br />
6. mDAB 1 _ m<br />
2 DB<br />
mDCB 1 _ m<br />
2 DB<br />
7. mDAB mDCB<br />
8. ADB CBD (AAS AAS)<br />
9. −−<br />
AB −−−<br />
CD (CPCTC)<br />
3 1. Circle O with diameters −−−−<br />
MOT and −−−−<br />
AOH<br />
2. MAT and HTA are right angles.<br />
3. −−−−<br />
MOT −−−−<br />
AOH<br />
4. −−<br />
AT −−<br />
AT<br />
5. MAT HTA (HL HL)<br />
6. −−−<br />
MA −−<br />
HT<br />
(CPCTC)<br />
4 1. Circle O, tangents PR , PV <br />
2. PR OR ; PV OV <br />
3. ORP and OVP are right angles.<br />
4. mORP mOVP<br />
5. −−−<br />
OR −−−<br />
OV<br />
6. −−<br />
PR −−<br />
PV<br />
7. RPO VPO (SAS SAS)<br />
8. RPO VPO (CPCTC)<br />
5 1. Circle O with T as <strong>the</strong> midpoint of CH<br />
2. m CT m TH<br />
3. m CTH 180<br />
4. m CT 90<br />
5. mCOT m CT<br />
6. COT and CTH are right angles.<br />
7. mCOT mCTH<br />
13-7 Circle Proofs 83
8. mHCT 1 _ m<br />
2 TH<br />
mCHT 1 _ CT<br />
2 m <br />
9. mHCT mCHT<br />
10. CTO HTC<br />
11. CT _<br />
CH<br />
TO<br />
_<br />
TH (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
6 1. Circle O with diameter −−−−<br />
DOG ; DR GI<br />
2. mTEG 1 _ m (<br />
2 DR TG )<br />
3. mTRI 1 _ m (<br />
2 GI TG )<br />
4. TG GI TGI<br />
5. mTEG 1 _ m<br />
2 TGI<br />
6. mTRI 1 _ m<br />
2 TGI<br />
7. mTEG mTRI<br />
8. mRTI mRTI<br />
9. TEX TRI<br />
10.<br />
(AA)<br />
TX _ <br />
EX<br />
TI _<br />
RI<br />
11. TX RI EX TI (The product of<br />
means equals<br />
<strong>the</strong> product of<br />
extremes.)<br />
7 1. CT CH<br />
2. −−−<br />
HA −−<br />
TD<br />
3. CTD CHA; TDH HAT<br />
4. TDH is supplementary to TDC;<br />
HAT is supplementary to HAC.<br />
5. TDC HAC<br />
6. TDC HAC<br />
7.<br />
(AAS AAS)<br />
−−<br />
CT −−−<br />
CH (CPCTC)<br />
8. HA TD<br />
8 1.<br />
(Contradiction)<br />
−−<br />
BC −−−<br />
QR<br />
2. −−<br />
BA −−<br />
QP<br />
3. BC RQ<br />
4. BA QP<br />
(Congruent chords<br />
subtend congruent<br />
arcs.)<br />
5. A D<br />
6. B Q<br />
(Inscribed angles<br />
intercepting congruent<br />
arcs are<br />
congruent.)<br />
7. RQP CBA<br />
8. AC PR<br />
(SAS SAS)<br />
84 Chapter 13: <strong>Geometry</strong> of <strong>the</strong> Circle<br />
9 a 1. BEA CED (Vertical angles are<br />
congruent.)<br />
2. BDC CAB (Congruent<br />
inscribed angles)<br />
3. BEA CED<br />
b (i) 4 : 5 (ii)<br />
(AA)<br />
16 _<br />
25<br />
10 1. −−<br />
BA −−−<br />
PGD<br />
2. DBA BDP<br />
3. mPBD 1 _ m<br />
2 BGD<br />
4. mBAD 1 _ m<br />
2 BGD<br />
5. mPBD mDAB<br />
6. PBD DAB<br />
7. PBD DAB<br />
8. PD _ <br />
BD<br />
BD _<br />
BA (Corresponding<br />
sides of similar<br />
triangles are in<br />
11 1.<br />
proportion.)<br />
−−<br />
AB is <strong>the</strong> diameter.<br />
2. Line l is tangent to circle O.<br />
3. ABC and AEB are right angles.<br />
4. ABC AEB<br />
5. CAB BAE<br />
6. ABC AEB (AA)<br />
7. AC _ <br />
AB<br />
AB _<br />
AE (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
12 1. −−<br />
AC is <strong>the</strong> diameter.<br />
2. −−<br />
BD intersects <strong>the</strong> diameter at point E.<br />
3. −−<br />
AB −−<br />
BE<br />
4. BAE BEA<br />
5. BEA CED<br />
6. BAE CED<br />
7. ABE and DCE are right angles.<br />
8. ABE DCE<br />
9. ABE ECD<br />
10.<br />
(AA)<br />
AC _ <br />
ED _<br />
AB<br />
EC<br />
11. AC EC (The product of<br />
ED AB means equals<br />
<strong>the</strong> product of<br />
extremes.)<br />
13 1. DCG CBD<br />
2. mE 1 _ m (<br />
2 ABC DF )<br />
1 _ m (<br />
2 CB )
3. mCDB 1 _ m (<br />
2 CB )<br />
4. mE mCDB<br />
5. CBD FCE (AA)<br />
6. EF _ EC<br />
<br />
CD _<br />
DB (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
14 1. AZT MZH<br />
2. HMT TAH<br />
3. MZH AZT (AA)<br />
4. MZ _ <br />
AZ<br />
ZH _<br />
ZT (Corresponding<br />
sides of similar<br />
triangles are in<br />
proportion.)<br />
15 a 1. BAC BDC<br />
2. ABD ACD<br />
(Congruent inscribed<br />
angles)<br />
3. ACD ABE<br />
b BE 6<br />
(AA)<br />
13-8 Circles in <strong>the</strong><br />
Coordinate Plane<br />
(pages 386–387)<br />
1 (2) (x 4) 2 (y 3) 2 36<br />
2 (4) (13, 1)<br />
3 a (x 1) 2 (y 4) 2 25<br />
b (x 3) 2 (y 2) 2 49<br />
c (x 4) 2 (y 1) 2 121<br />
d (x 2) 2 (y 5) 2 64<br />
e (x 2) 2 y 2 121<br />
f x 2 y 2 144<br />
4 a x 2 y 2 49<br />
b (x 4) 2 y 2 25<br />
c x 2 y 2 41<br />
d (x 2) 2 y 2 36<br />
e (x 3) 2 (y 2) 2 9<br />
f (x 1) 2 (y 3) 2 25<br />
5 For parts a–d, check students’ graphs. The<br />
center and radius of each circle are listed<br />
below.<br />
a C(2, 4), r 3<br />
b C(3, 4), r 5<br />
c C(0, 2), r 4<br />
d C(6, 0), r 2<br />
6 a C(0, 0), r 9<br />
b C(0, 0), r 11<br />
c C(0, 0), r 20<br />
d C(0, 0), r 1<br />
e C(0, 0), r √ 2 <br />
7 Note: Students’ answers may vary <strong>for</strong> <strong>the</strong><br />
two o<strong>the</strong>r points on <strong>the</strong> circle.<br />
a C(5, 3), r 8; (5, 11), (5, 5)<br />
b C(4, 0), r 10; (6, 0), (14, 0)<br />
c C(2, 5), r 4; (2, 5), (6, 5)<br />
d C(3, 7), r 2 √ 3 ; (3, 7 2 √ 3 ),<br />
(3, 7 2 √ 3 )<br />
8 a x 2 y 2 25<br />
b x 2 y 2 4<br />
c x 2 y 2 34<br />
d x 2 y 2 37<br />
e x 2 y 2 68<br />
9 a (x 2) 2 (y 3) 2 2<br />
b (x 3) 2 (y 7) 2 80<br />
10 a x 2 (y 0.5) 2 12.25<br />
b (x 3) 2 (y 3) 2 5<br />
11 a Circumference 8, area 16<br />
b Circumference 12, area 36<br />
c Circumference 2 √ 13 , area 13<br />
12 a (3, 0)<br />
b (x 3) 2 (y 1) 2 1<br />
c <br />
13 a C(1, 3), r 2 √ 2 <br />
b Area 8, circumference 4 √ 2 <br />
13-9 Tangents, Secants,<br />
and <strong>the</strong> Circle in <strong>the</strong><br />
Coordinate Plane<br />
(pages 392–393)<br />
1 (1) 0<br />
2 a (3, 11), (3, 1)<br />
b (3) x 1<br />
3 y 0<br />
4 y 0.5x 2.5<br />
5 y x 4<br />
6 y 3<br />
7 y 4<br />
8 y 2x 10<br />
9 a (4, 3), (3, 4) b secant<br />
10 a (0, 4), (4, 0) b secant<br />
11 a (2, 3), (2, 5) b secant<br />
12 a (0, 3) b tangent<br />
13 a (0, 2), (2, 0) b secant<br />
14 a (5, 5) b tangent<br />
15 a (1, 4), (1, 4) b secant<br />
13-9 Tangents, Secants, and <strong>the</strong> Circle in <strong>the</strong> Coordinate Plane 85
16 a (2 √ 2 , 2 √ 2 ), (2 √ 2 , 2 √ 2 ) b secant<br />
17 a (10, 0) and (10, 0) b secant<br />
18 a (7, 7) and (7, 7) b secant<br />
19 a (6, 0) b tangent<br />
20 a (8, 6) and (6, 4) b secant<br />
21 a (0, 5) and (4, 3) b secant<br />
22 a Sub-in <strong>the</strong> given points in (x 3) 2 <br />
(y 2) 2 25.<br />
b Midpoint of −−−<br />
ME is (3.5, 2.5);<br />
distance √ 0.5 .<br />
23 a y x 8<br />
b y x 8<br />
c (0, 8)<br />
24 a y 3x 10 b x 10 c P(10, 20)<br />
d PA √ 160 4 √ 10 , PB √ 1000 <br />
10 √ 10 , PM 20<br />
PA PB PM 2<br />
4 √ 10 10 √ 10 20 2<br />
400 400<br />
Chapter Review (page 393–397)<br />
1 (1) All chords in a circle are congruent.<br />
2 (2) (x 4) 2 (y 2) 2 9<br />
3 (2) 1<br />
4 (3) 8<br />
5 (3) 6 and 15<br />
6 (2) 12<br />
7 (1) 12<br />
8 (2) 6<br />
9 (2) 68<br />
10 (3) 122<br />
11 (1) 43<br />
12 (3) 86<br />
13 (2) 2 : 1<br />
14 (1) 41<br />
15 (3) 10 inches<br />
16 (3) 125<br />
17 (1) 40<br />
18 (4) 100<br />
19 (2) 230<br />
20 16<br />
21 90<br />
22 104<br />
23 20<br />
24 48<br />
25 12<br />
26 a 40 b 40 c 40 d 40<br />
e 110 f 110 g 35<br />
86 Chapter 13: <strong>Geometry</strong> of <strong>the</strong> Circle<br />
27 a 40 b 70 c 20<br />
d 45 e 25 f 65<br />
28 a 30 b 45 c 15<br />
d 135 e 90<br />
29 a C(0, 0), r 20<br />
b Answers may vary. (0, 20), (0, 20)<br />
30 a C(7, 11), r 9<br />
b Answers may vary. (7, 20), (7, 2)<br />
31 a C(3, 13), r 6<br />
b Answers may vary. (3, 19), (3, 7)<br />
32 a C(3, 5), r 2 √ 5<br />
b Answers may vary. (3, 5 2 √ 5 ),<br />
(3, 5 2 √ 5 )<br />
33 (x 3) 2 (y 2) 2 16, C(3, 2), r 4<br />
34 a (x 4) 2 (y 2) 2 49<br />
b (x 5) 2 y 2 2<br />
c x 2 y 2 169<br />
d C(3, 0), r 5, (x 3) 2 y 2 25<br />
e C(4, 4) r √ 10 , (x 4) 2 (y 4) 2 10<br />
2 ) 2<br />
35 6 2 6 2 (6 √ <br />
36 y 1 _<br />
x 10<br />
3<br />
37 a (4, 3) and (4, 3) b secant<br />
38 a (0, 3) and (3, 0) b secant<br />
39 a (8, 2) and (2, 8) b secant<br />
40 a (0, 5) and (3, 4) b secant<br />
Note: Since <strong>the</strong>re are many variations of proofs,<br />
<strong>the</strong> following is simply one set of acceptable<br />
statements to complete each proof. Depending<br />
on <strong>the</strong> textbook used, <strong>the</strong> wording and <strong>for</strong>mat<br />
of reasons may differ, so <strong>the</strong>y have not been<br />
supplied <strong>for</strong> <strong>the</strong> method of congruence applied<br />
in each problem. (These solutions are intended<br />
to be used as a guide—o<strong>the</strong>r possible solutions<br />
may vary.)<br />
41 1. AOB COB<br />
2. −−<br />
AB −−<br />
CB<br />
3. AB CB<br />
42 1. −−−<br />
DA is <strong>the</strong> diameter of circle O.<br />
2. Radii −−−<br />
OA , −−−<br />
OC , and −−−<br />
OD<br />
3. −−−<br />
OA −−−<br />
OC −−−<br />
OD (Radii are equal.)<br />
4. −−−<br />
AD −−−<br />
CD<br />
5. AOD COD (SSS SSS)<br />
6. AD CD<br />
7. ABD CBD (Congruent arcs<br />
are intercepted<br />
by congruent inscribed<br />
angles.)
43 1. −−−<br />
MA −−−<br />
OK , −−−<br />
MD −−<br />
OL<br />
2. OAM and ODM are right angles.<br />
3. −−−<br />
AM −−−<br />
DM<br />
4. −−−<br />
OM −−−<br />
OM<br />
5. OAM ODM (HL HL)<br />
6. AOM LOM (CPCTC)<br />
7. MK ML (Congruent angles<br />
have congruent<br />
arcs.)<br />
Locus and<br />
Constructions<br />
14-1 Basic Constructions<br />
(page 404)<br />
Note: For exercises 1–20, check students’ constructions;<br />
procedures may vary.<br />
1 ab Use construction of congruent angles<br />
procedure.<br />
2 Use construction of a perpendicular to a line<br />
through a given point on <strong>the</strong> line procedure.<br />
3 ac Use construction of an angle bisector<br />
procedure.<br />
4 Use construction of a line perpendicular to<br />
a line through a given point not on <strong>the</strong> line<br />
procedure.<br />
5 Use construction of a perpendicular bisector<br />
of a line segment procedure.<br />
6 Use construction of a parallel line through a<br />
given point not on <strong>the</strong> line procedure.<br />
7 Use construction of <strong>the</strong> median of a triangle<br />
procedure.<br />
8 Use construction of line perpendicular to a<br />
line through a given point not on <strong>the</strong> line<br />
procedure.<br />
44 1. −−<br />
OE −−<br />
AB , OD −−<br />
AC<br />
2. AEO and ADO are right angles.<br />
3. −−<br />
AB −−<br />
AC<br />
4. −−<br />
OE bisects −−<br />
AB .<br />
5. −−−<br />
OD bisects −−<br />
AC .<br />
6. −−<br />
AE −−−<br />
AD<br />
7. −−−<br />
OA −−−<br />
OA<br />
8. AEO ADO (HL HL)<br />
45 Use equal arcs are subtended by equal<br />
chords.<br />
CHAPTER<br />
14<br />
9 Use construction of a perpendicular bisector<br />
procedure.<br />
10 Use construction of an angle bisector<br />
procedure.<br />
11 a Use construction of a perpendicular line<br />
procedure.<br />
b Use construction of a perpendicular line<br />
procedure and <strong>the</strong>n construct an angle<br />
bisector.<br />
12 a Use construction of an equilateral triangle<br />
procedure.<br />
b Using angle constructed in a, construct an<br />
angle bisector.<br />
c Construct angle bisector of b.<br />
13 Using <strong>the</strong> vertex of <strong>the</strong> angle as <strong>the</strong> center<br />
of a circle and one of <strong>the</strong> legs as its radius,<br />
construct a circle. Construct <strong>the</strong> diameter by<br />
extending <strong>the</strong> radius.<br />
14 Using <strong>the</strong> vertex of <strong>the</strong> angle as <strong>the</strong> center<br />
of a circle and one of <strong>the</strong> legs as its radius,<br />
construct a circle. Construct <strong>the</strong> diameter by<br />
extending <strong>the</strong> radius. Construct a perpendicular<br />
through <strong>the</strong> center of <strong>the</strong> circle.<br />
14-1 Basic Constructions 87
15 a Use constructing a congruent angle<br />
procedure.<br />
b Use constructing a congruent angle<br />
procedure.<br />
c Construct a perpendicular bisector of a<br />
segment to <strong>for</strong>m a 90 angle. Then use<br />
constructing a congruent angle procedure<br />
to construct <strong>the</strong> sum.<br />
d Use constructing an angle bisector<br />
procedure.<br />
e Use constructing an angle bisector<br />
procedure <strong>for</strong> A. Then use constructing a<br />
congruent angle procedure to construct<br />
<strong>the</strong> sum.<br />
16 Use construction of an equilateral triangle<br />
procedure.<br />
17 Use <strong>the</strong> construction of an equilateral<br />
triangle procedure using <strong>the</strong> length of <strong>the</strong><br />
longer segment to construct <strong>the</strong> leg of <strong>the</strong><br />
isosceles triangle.<br />
18 Using <strong>the</strong> compass, measure <strong>the</strong> radius and<br />
construct a circle passing through <strong>the</strong> point<br />
on <strong>the</strong> tangent line.<br />
19 Use construction of a line tangent to a given<br />
circle through a given point outside <strong>the</strong> circle<br />
procedure.<br />
20 Use procedures <strong>for</strong> constructing parallel lines<br />
and perpendicular lines.<br />
14-2 Concurrent Lines and<br />
Points of Concurrency<br />
(pages 407–408)<br />
1 (4) obtuse<br />
2 (3) at one of <strong>the</strong> vertices of <strong>the</strong> triangle<br />
3 Check students’ constructions.<br />
4 All at <strong>the</strong> same point<br />
5 10<br />
6 6 2 _<br />
3<br />
7 36<br />
8 16.5<br />
9 4.5<br />
10 SP 4, SC 6<br />
11 PB 15, PR 30<br />
12 PA 15, QA 45<br />
13 4 1 _<br />
3<br />
14 9<br />
88 Chapter 14: Locus and Constructions<br />
15 3 √ 3 <br />
16 QE 4, DE 6<br />
17 a, b, c: all interior<br />
18 a, b, c: all interior<br />
19 a interior b on a side c exterior<br />
20 a interior b at a vertex c exterior<br />
21 Incenter: √ 3 , circumcenter: 2 √ 3<br />
22 Incenter: 2 √ 3 , circumcenter: 4 √ 3 <br />
23 Incenter: 3 √ 3 , circumcenter: 6 √ 3 <br />
24 Incenter: 6, circumcenter: 12<br />
25 Since AG GC, <strong>the</strong> base is <strong>the</strong> same and <strong>the</strong><br />
altitude is <strong>the</strong> same. There<strong>for</strong>e, <strong>the</strong> area is <strong>the</strong><br />
same.<br />
14-4 Six Fundamental Loci<br />
and <strong>the</strong> Coordinate Plane<br />
(pages 413–414)<br />
1 (3) One concentric circle of radius 11 inches<br />
2 (2) one line<br />
3 (2) a point<br />
4 (1) a circle of radius 4 with center at P<br />
5 (4) a circle<br />
Note: For exercises 6–17 check students’ sketches.<br />
6 Circle with given point as center and<br />
radius 3<br />
7 Two parallel lines, one on each side of <strong>the</strong><br />
given line<br />
8 One line parallel to and midway between <strong>the</strong><br />
given parallel lines<br />
9 The perpendicular bisector of <strong>the</strong> segment<br />
joining R and S<br />
10 The perpendicular bisector of <strong>the</strong> segment<br />
AB<br />
11 A circle with radius 2.5 inches and <strong>the</strong> given<br />
point as <strong>the</strong> center<br />
12 The line that is <strong>the</strong> bisector of ABC<br />
13 Two lines each <strong>the</strong> bisector of <strong>the</strong> vertical<br />
angles<br />
14 Two concentric circles with radii 5 and 9<br />
15 One concentric circle midway between <strong>the</strong><br />
given circles<br />
16 All <strong>the</strong> points in <strong>the</strong> interior of a circle with<br />
<strong>the</strong> given point as <strong>the</strong> center and with a<br />
radius 2 inches<br />
17 A circle with <strong>the</strong> given point as <strong>the</strong> center<br />
and a radius of 3 inches and all <strong>the</strong> points in<br />
<strong>the</strong> exterior of that circle
18 Two lines, one on each side, parallel to <strong>the</strong><br />
given line and at a distance r from <strong>the</strong> given<br />
line<br />
19 One line parallel to <strong>the</strong> two parallel lines and<br />
midway between <strong>the</strong>m<br />
20 The diameter of <strong>the</strong> circle that is <strong>the</strong> perpendicular<br />
bisector of <strong>the</strong> given chord<br />
21 The diameter of <strong>the</strong> circle that is <strong>the</strong> perpendicular<br />
bisector of <strong>the</strong> given chord<br />
22 A ray that is <strong>the</strong> angle bisector<br />
23 A line perpendicular to <strong>the</strong> point on <strong>the</strong><br />
given line<br />
24 The perpendicular bisector of <strong>the</strong> chord that<br />
joins <strong>the</strong> two given points<br />
Locus in <strong>the</strong> Coordinate Plane<br />
(Pages 418–419)<br />
1 (4) y 2 or y 2<br />
2 (1) x 2<br />
3 (2) x 5<br />
4 (2) y 1<br />
5 (4) (x 1) 2 (y 3) 2 25<br />
6 (3) y 3 and x 0<br />
7 (4) y x 1 and y x 5<br />
8 y 1<br />
9 y 2, y 10<br />
10 y 4<br />
11 y x 3 and y x 3<br />
12 y x 3<br />
13 x 1 and y 0<br />
14 a (x 1) 2 (y 4) 2 16<br />
b x 2 y 2 36<br />
c x 2 (y 1) 2 9<br />
d (x 1) 2 y 2 1<br />
e x 2 (y 2) 2 6.25<br />
f (x 1) 2 (y 3) 2 30.25<br />
15 The showers could be placed anywhere on<br />
<strong>the</strong> perpendicular bisector of <strong>the</strong> line segment<br />
joining <strong>the</strong> diving boards, which is<br />
35 ft from each of <strong>the</strong> diving boards.<br />
14-5 Compound Locus and<br />
<strong>the</strong> Coordinate Plane<br />
(pages 421–422)<br />
1 (3) a pair of points<br />
2 (4) <strong>the</strong> empty set<br />
3 (1) 0<br />
4 (3) a pair of points<br />
5 (3) 2<br />
6 (2) 1<br />
7 a 4 b 2 c 0<br />
8 Find <strong>the</strong> point of concurrency (circumcenter)<br />
of <strong>the</strong> perpendicular bisectors of <strong>the</strong> sides of<br />
<strong>the</strong> triangle <strong>for</strong>med.<br />
9 3<br />
10 2<br />
11 Parallel lines 2 inches from <strong>the</strong> given line,<br />
one on each side<br />
a Circle with radius 3, center R. 1 point<br />
b Circle with radius 6, center R. 2 points<br />
c Circle with radius 7, center R. 3 points<br />
d Circle with radius 9, center R. 3 points<br />
12 For all parts, check students’ sketches.<br />
a Two intersecting circles, but not tangent.<br />
2 points<br />
b Two tangent circles. 1 point<br />
c Two disjoint circles. 0 points<br />
13 Sketch circle and two lines; 4 points.<br />
14 Sketch a line and a circle; 2 points.<br />
15 Sketch two lines bisecting vertical angles,<br />
and two parallel lines; 4 points.<br />
16 Sketch two concentric circles and two lines<br />
bisecting vertical angles; 8 points.<br />
17 The intersection of <strong>the</strong> line parallel to m and<br />
k and midway between <strong>the</strong>m, and a circle<br />
with A as <strong>the</strong> center and d as <strong>the</strong> radius.<br />
Case I d 1 _ f (0 points)<br />
2<br />
Case II d 1 _ f (1 point)<br />
2<br />
Case III d 1 _ f (2 points)<br />
2<br />
18 The intersection of <strong>the</strong> perpendicular<br />
bisector of segment AB and a circle about<br />
center A with radius x.<br />
Case I Radius x 1 _ d (0 points)<br />
2<br />
Case II Radius x 1 _ d (1 point)<br />
2<br />
Case III Radius x 1 _ d (2 points)<br />
2<br />
Compound Loci and <strong>the</strong> Coordinate Plane<br />
(pages 423–424)<br />
1 (4) 4<br />
2 (3) 2<br />
3 (4) 4<br />
4 a 2 b 1 c 0<br />
14-5 Compound Locus and <strong>the</strong> Coorinate Plane 89
5 Two horizontal lines y 11 and y 1, and<br />
vertical line x 4. Locus: 2 points.<br />
6 Locus 2 points: (0, 8) and (8, 0)<br />
7 Locus 2 points: (2, 7) and (2, 1)<br />
8 Locus 2 points: (3, 1) and (2, 2)<br />
9 a y 2 b x 4 c (4, 2)<br />
d (x 2) 2 (y 2) 2 16 e one<br />
10 a Circle with radius, d<br />
b Two lines: x 1 and x 1<br />
c (i) 1 point (ii) 3 points (iii) 4 points<br />
14-6 Locus of Points<br />
Equidistant From a Point<br />
and a Line<br />
(page 427)<br />
Note: For exercises 1–10, check students’<br />
sketches.<br />
1 b (0, 2), (2, 3), (2, 3)<br />
x 2<br />
c y _ 2<br />
4<br />
2 b (2, 1), (2, 1), (6, 1)<br />
2<br />
c y _ x<br />
<br />
x<br />
8 _ <br />
2 1 _<br />
2<br />
3 b (2, 0), (2, 2), (6, 2)<br />
2<br />
c y <br />
x<br />
<br />
_ <br />
1<br />
8 _ x <br />
2 1 _<br />
2<br />
4 b (3, 0.5), (0, 1), (6, 1)<br />
x 2<br />
c y _ x 1<br />
6<br />
5 b (1, 0), (3, 4), (3, 4)<br />
y 2<br />
c x _ 1<br />
8<br />
6 b (3, 1.5), (6, 0), (0, 0)<br />
x 2<br />
c y <br />
_ x<br />
6<br />
7 b (2, 2), (6, 4), (2, 4)<br />
x 2<br />
c y <br />
_ <br />
1<br />
8 _ x <br />
2 5 _<br />
2<br />
8 b (1, 4), (4, 10), (4, 2)<br />
2<br />
y<br />
c x _<br />
2y<br />
<br />
12 _ <br />
7<br />
3 _<br />
3<br />
9 b (0, 0), (2, 4), (2, 4)<br />
y 2<br />
c x _<br />
8<br />
10 b (3, 1.5), (1, 2), (6, 2)<br />
x 2<br />
c y _ x 2<br />
6<br />
90 Chapter 14: Locus and Constructions<br />
14-7 Solving O<strong>the</strong>r Linear-<br />
Quadratic and Quadratic-<br />
Quadratic Systems<br />
(page 430)<br />
Note: For exercises 1–20, check students’<br />
sketches.<br />
1 (2, 0), (2, 0)<br />
2 (2 √ 2 , 0), (2 √ 2 , 0)<br />
3 (0, 5), (4, 3)<br />
4 (3, 2)<br />
5 (2, 3), (2, 5)<br />
6 (2, 1), (1, 2)<br />
7 (1, 1), (1, 1)<br />
8 (5, 27), (1, 5)<br />
9 ( 5 _ , 6) , (2, 5)<br />
3<br />
10 (4, 0), (4, 0)<br />
11 ( √ 2 , 0), ( √ 2 , 0)<br />
12 (3, 2), (6, 1)<br />
13 (2 √ 2 , √ 2 ), (2 √ 2 , √ 2 ), ( √ 2 , 2 √ 2 ),<br />
( √ 2 , 2 √ 2 )<br />
14 (0.5, 1.25), (4, 3)<br />
15 (1, 0)<br />
16 (1, 3), (1, 3)<br />
17 (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ), (2 √ 2 , √ 5 ),<br />
(2 √ 2 , √ 5 )<br />
18 (2, 1), (2, 1)<br />
19 (3, 0)<br />
20 (2, 4), (2, 0)<br />
Chapter Review (pages 430–433)<br />
1 (2) acute triangles<br />
2 (4) median to side −−<br />
AC<br />
3 (2) AAS<br />
4 (2) two circles<br />
5 (2) X lies on <strong>the</strong> locus of points equidistant<br />
from R and S.<br />
6 (2) y 2x 3<br />
7 (3) (x 3) 2 (y 4) 2 36<br />
8 (1) x 3<br />
9 (3) x 5<br />
10 (1) (0, 0) and (0, 8)<br />
11 (4) (0, 2) and (4, 2)<br />
12 (3) perpendicular bisectors of <strong>the</strong> sides of <strong>the</strong><br />
triangle<br />
13 (3) 4<br />
14 (3) 3<br />
15 (3) 2
16 (4) 4<br />
17 (1) 90 x<br />
18 8<br />
19 x 3, PD 6, RP 12, RD 18<br />
20 x 2, y 4, BE 9, AD 12<br />
Note: For exercises 21–24, check students’<br />
sketches.<br />
21 Two concentric circles. Locus: radius 1 and<br />
radius 7.<br />
22 Circle, radius 3, x 2 y 2 9. Two lines,<br />
x 4, x 4. Locus: 0 points<br />
23 Circle: x 2 y 2 4. Circle: x 2 (y 4) 2 9.<br />
Locus: 1 point, (0, 2)<br />
24 Two lines parallel to line m on opposite sides<br />
and 4 centimeters from m. Circle with center<br />
H and radius 2. Locus: 1 point<br />
25 (x 4) 2 (y 2) 2 1<br />
26 y x 4<br />
27 Two lines: y x 2 and y x 2<br />
Note: For exercises 28–37, check students’<br />
sketches.<br />
28 b (2, 0.5), (5, 1), (1, 1)<br />
2<br />
c y _ x<br />
<br />
2x<br />
6 _ <br />
1<br />
3 _<br />
6<br />
29 b (5, 7), (1, 3), (3, 1)<br />
2<br />
y<br />
c x _<br />
3y<br />
<br />
8 _ <br />
1<br />
4 _<br />
8<br />
30 b (2, 0), (2, 2), (6, 2)<br />
x 2<br />
c y <br />
_ <br />
x<br />
8 _ <br />
2 1 _<br />
2<br />
31 b (1, 0), (1, 4), (1, 4)<br />
y 2<br />
c x _ 1<br />
8<br />
32 (0, 5), (3, 4), (3, 4)<br />
33 (0, 5), (3, 8)<br />
34 (0, 3), (3, 9)<br />
35 (2 √ 7 , 1 √ 7 ), (2 <br />
36 (3, 10), (1, 6)<br />
√ 7 , 1 √ 7 )<br />
37 (0, 4), ( √ 7 , 3), ( √ 7 , 3)<br />
38 The intersection of <strong>the</strong> perpendicular bisector<br />
of AB, and circle with A as center<br />
Case I x 1 _ f (0 points)<br />
2<br />
Case II x 1 _ f (1 point)<br />
2<br />
Case III x 1 _ f (2 points)<br />
2<br />
39 The intersection of one circle with radius of<br />
2.5, concentric with <strong>the</strong> given circles, and<br />
two lines parallel to <strong>the</strong> given line, one on<br />
each side at a distance x from <strong>the</strong> given line.<br />
Case I x 2.5 (4 points)<br />
Case II x 2.5 (2 points)<br />
Case III x 2.5 (0 points)<br />
40 The intersection of circle with center P<br />
and radius d and circle with center Q and<br />
radius d.<br />
Case I x 2d (2 points)<br />
Case II x 2d (1 point)<br />
Case III x 2d (0 points)<br />
41 The intersection of two lines parallel to line<br />
m, one on each side at a distance d from line<br />
m and <strong>the</strong> circle with center A and radius r.<br />
Case I r d (0 points)<br />
Case II r d (2 points)<br />
Case III r d (4 points)<br />
42 Locus: A line parallel to <strong>the</strong> north side and<br />
south side of <strong>the</strong> courtyard and midway<br />
between <strong>the</strong>m. Since <strong>the</strong> width of <strong>the</strong> garden<br />
is 60 ft, every point on this line will be equidistant<br />
from <strong>the</strong> north and south walls and<br />
at least 30 feet from <strong>the</strong> North Entrance.<br />
43 Check students’ constructions of <strong>the</strong> orthocenter<br />
of an acute triangle.<br />
44 Check students’ constructions of <strong>the</strong> perpendicular<br />
bisector of <strong>the</strong> three sides of a right<br />
triangle.<br />
45 Check students’ constructions of a circle that<br />
passes through <strong>the</strong> three vertices of an obtuse<br />
triangle.<br />
Chapter Review 91
Each review has a total of 58 possible points. Use <strong>the</strong> following table, adapted from <strong>the</strong> <strong>Regents</strong><br />
<strong>Examination</strong>s, to convert <strong>the</strong> student’s raw score to a scaled score.<br />
Raw Score Scaled Score Raw Score Scaled Score Raw Score Scaled Score<br />
58 100 38 75 19 49<br />
57 99 37 74 18 47<br />
56 98 36 72 17 46<br />
55 97 35 70 16 45<br />
54 96 34 68 15 44<br />
53 93 33 67 14 43<br />
52 92 32 66 13 42<br />
51 91 31 65 12 41<br />
50 90 30 64 11 39<br />
49 89 29 63 10 37<br />
48 87 28 61 9 36<br />
47 86 27 60 8 34<br />
46 85 26 59 7 32<br />
45 84 25 58 6 29<br />
44 82 24 57 5 26<br />
43 80 23 56 4 21<br />
42 79 22 54 3 16<br />
41 78 21 53 2 10<br />
40 77 20 51 1 5<br />
39 76<br />
Chapters 1–2<br />
(pages 434–437)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (4) a → r 2 (2) −−<br />
AB 3 (3) p q<br />
92<br />
Cumulative Reviews
4 (4) 2BC AC<br />
5 (1) If I win a scholarship, <strong>the</strong>n I will play soccer.<br />
6 (3) 2 _<br />
3<br />
7 (3) The triangle may be acute.<br />
8 (4) If two angles are not congruent, <strong>the</strong>n <strong>the</strong>y are not right angles.<br />
9 (2) right<br />
10 (1) isosceles<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 AB CD EF, and an appropriate explanation is given.<br />
1 AB CD EF, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 3, and an appropriate explanation is given.<br />
1 3, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 55, and an appropriate explanation is given.<br />
1 55, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 9, and an appropriate explanation is given.<br />
1 9, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–2 93
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 a (i) scalene (ii) right triangle<br />
94 Cumulative Reviews<br />
b (i) isosceles (ii) acute<br />
c (i) equilateral (ii) equiangular<br />
d (i) isosceles (ii) obtuse<br />
e (i) isosceles (ii) right triangle<br />
f (i) scalene (ii) obtuse<br />
3 Answered all but two parts correctly.<br />
2 Answered all but three parts correctly.<br />
1 Answered two parts correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 3 : 1, and an appropriate explanation is given.<br />
3 An appropriate method is shown, but AB _ is given instead.<br />
BD<br />
2 An appropriate method is shown, but an incorrect answer is given.<br />
1 3 : 1, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 x 77, and an appropriate method is used.<br />
3 An appropriate method is shown, but y is given instead.<br />
2 An appropriate method is shown, but an incorrect answer is given.<br />
1 x 77, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 a distributive property<br />
b associative property of addition<br />
c additive identity<br />
d commutative property of multiplication<br />
5 Student answers all but one part correctly.<br />
3 Student answers only two parts correctly.<br />
1 Student answers only one part correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
6 34, and an appropriate method is given.<br />
5 An appropriate method is used, but a single arithmetical error is made.<br />
4 An appropriate method is used, but two arithmetical errors are made.<br />
3 An appropriate method is used, but student found m1 or m2.<br />
2 An appropriate method is shown, but multiple computational mistakes are made.<br />
1 34, but no appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
6 a {6, 7, 8, 9, 10}<br />
b {0, 2, 4, 6, 8, 10}<br />
c {1, 3, 5}<br />
d {0, 1, 2, 3, 4, 5, 7, 9}<br />
5 One part is missing elements from <strong>the</strong> solution set.<br />
4 Elements are missing from two solution sets.<br />
3 Answered two parts correctly.<br />
2 Answered all parts, but solution sets are missing elements.<br />
1 Only one part is answered correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–2 95
Chapters 1–3<br />
(pages 438–441)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) 80<br />
2 (2) 54<br />
3 (1) Subtraction postulate<br />
4 (4) If Cecilia is not a senior, <strong>the</strong>n she does not take AP Calculus.<br />
5 (4) 22<br />
6 (3) If a triangle is not equilateral, <strong>the</strong>n it is not isosceles.<br />
7 (1) 20<br />
8 (3) 1 _<br />
2<br />
9 (4) 3 _<br />
2<br />
10 (2) b c<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 70, and an appropriate explanation is given.<br />
1 70, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 Affirming <strong>the</strong> conclusion does not affirm <strong>the</strong> premise.<br />
a The quadrilateral could be rhombus or any o<strong>the</strong>r figure.<br />
b x could be 7 or any o<strong>the</strong>r positive number less than 7, so that x is not greater than 8.<br />
1 Student answers only one part correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 mr 20, and an appropriate explanation is given.<br />
1 mr 20, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 RB 2, and an appropriate explanation is given.<br />
1 RB 2, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
96 Cumulative Reviews
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 mA 82, mB 82, and mC 16, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used to find x, but <strong>the</strong> measures of <strong>the</strong> angles are not given.<br />
1 mA 82, mB 82, and mC 16, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 a ABD and DBE<br />
b ABE and EBC<br />
c DBC<br />
d ABC<br />
3 Student answers all but one part correctly.<br />
2 Student answers only two parts correctly.<br />
1 Student answers only one part correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 m4 50, m5 20, mABD 160, and an appropriate method is used.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 m4 50, m5 20, but student did not give <strong>the</strong> measure of ABD.<br />
1 m4 50, m5 20, mABD 160, but no appropriate method is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–3 97
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 a If <strong>the</strong> sum of <strong>the</strong> measures of two acute angles is 90, <strong>the</strong> two angles are<br />
complementary.<br />
b If two acute angles are not complementary, <strong>the</strong>n <strong>the</strong>ir sum is not 90.<br />
c If <strong>the</strong> sum of <strong>the</strong> measures of two acute angles is not 90, <strong>the</strong>n <strong>the</strong> two acute angles<br />
are not complementary.<br />
d Two acute angles are complementary if and only if <strong>the</strong> sum of <strong>the</strong>ir measures is 90.<br />
5 Student answers all but one part correctly.<br />
3 Student answers only two parts correctly.<br />
1 Student answers only one part correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
1 (2) Reflexive postulate<br />
2 (3) Addition postulate<br />
1 (4) Partition postulate<br />
2 (5) Substitution postulate<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
1 (2) Definition of bisector<br />
1 (3) Definition of midpoint<br />
2 (4) Doubles of equals are equal.<br />
2 (5) Substitution postulate<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–4<br />
(pages 442–445) Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) 220 5 (2) 120<br />
2 (2) a median 6 (1) 3 and 2 are nonadjacent complementary angles.<br />
3 (4) 120 7 (2) b<br />
4 (3) If <strong>the</strong> diagonals of a quadrilateral 8 (3) BAR and RAI<br />
are not congruent, <strong>the</strong>n <strong>the</strong> 9 (1) mx my<br />
quadrilateral is not a rectangle. 10 (3) a b 4<br />
98 Cumulative Reviews
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 a True b False c False d True<br />
1 Student answers only two parts correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 11, and an appropriate explanation is given.<br />
1 11, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 68 and 112, and an appropriate explanation is given.<br />
1 68 and 112, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 57.5, and an appropriate explanation is given.<br />
1 57.5, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 AC 12, AB 16, BC 16, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 AC 12, AB 16, BC 16, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–4 99
16 Score Explanation<br />
4 a 91, b 35, c 54, d 91, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 a 91, b 35, c 54, d 91, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 mPTC 50, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 mPTC 50, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
3 a Given A F and −−<br />
AB −−<br />
EF . −−−<br />
AD −−<br />
CF and by <strong>the</strong> reflexive property of<br />
congruence −−−<br />
DC −−−<br />
DC . By <strong>the</strong> addition postulate, −−−−<br />
ADC −−−<br />
FCD . There<strong>for</strong>e, by SAS,<br />
ABC DEF.<br />
2 a An appropriate proof with correct conclusions is shown, but one reason is faulty<br />
and/or one statement is missing.<br />
1 a A correct conclusion is reached and a reason is given, but no appropriate method<br />
is shown.<br />
3 b Given D A. C is <strong>the</strong> midpoint of −−−<br />
AD , <strong>the</strong>n −−<br />
AC −−−<br />
DC .<br />
ACB DCE because vertical angles are congruent. There<strong>for</strong>e,<br />
by ASA, ABC DEC.<br />
2 b An appropriate proof with correct conclusions is shown, but one reason is faulty<br />
and/or one statement is missing.<br />
1 b A correct conclusion is reached and a reason is given, but no appropriate method<br />
is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
100 Cumulative Reviews
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. B D 1. Given.<br />
2. −−<br />
BC −−−<br />
DC 2. Given.<br />
3. BCA DCE 3. Vertical angles are congruent.<br />
4. ACB ECD 4. ASA ASA.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 ACB ECD and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
BD is <strong>the</strong> median to −−<br />
AC . 1. Given.<br />
2. D is <strong>the</strong> midpoint of −−<br />
AC . 2. Definition of a median of an isosceles triangle.<br />
3. −−−<br />
AD −−−<br />
DC 3. Definition of a midpoint.<br />
4. −−<br />
AB −−<br />
BC 4. Given.<br />
5. −−<br />
BD −−<br />
BD 5. Reflexive Property of Congruence.<br />
6. I II 6. SSS SSS.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or<br />
three steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than three reasons are<br />
faulty or more than three steps are missing or have errors.<br />
1 I II and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–4 101
Chapters 1–5<br />
(pages 446–449)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (4) −−<br />
QB , −−−<br />
QA , ____<br />
QM 6 (1) isosceles<br />
2 (1) q → p 7 (3) −−−<br />
CD bisects −−<br />
AB .<br />
3 (3) 70 8 (2) 3 4<br />
4 (2) 9 (3) 28<br />
5 (4) ADB WDB 10 (3) 78.5<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 6, and an appropriate explanation is given.<br />
1 6, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 x 8, A 81, B 99, and an appropriate explanation is given.<br />
1 x 8, A 81, B 99, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 256, and an appropriate explanation is given.<br />
1 256, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
1 a 45, and an appropriate explanation is given.<br />
1 b 90, and an appropriate explanation is given.<br />
1 (a) 45 and (b) 90, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
102 Cumulative Reviews
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
2 a BCA ACD<br />
2 b −−−<br />
AD −−<br />
AB<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
DB −−<br />
AC , −−−<br />
AD −−−<br />
DC 1. Given.<br />
2. DBA and ADC are right<br />
angles.<br />
3. 1 is complementary to x. 3. Given.<br />
2. Definition of perpendicular lines.<br />
4. m1 mx 90 4. Definition of complementary angles.<br />
5. mx m2 mADC 5. Addition postulate.<br />
6. mx m2 90 6. Substitution postulate.<br />
7. m1 mx mx m2 7. Substitution postulate.<br />
8. m1 m2 8. Subtraction postulate.<br />
9. 1 2 9. Angles with equal measures are congruent.<br />
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 1 2 and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
2 a Since −−−<br />
CD bisects −−<br />
AB at E, −−<br />
AE −−<br />
BE . Given 1 2, and because vertical angles are<br />
congruent, AEC BED. Hence, by ASA, ACE BDE.<br />
1 a A correct conclusion is reached and reason is given, but no appropriate method<br />
is shown.<br />
2 b Given 1 2 and −−<br />
LN −−−<br />
MO . LNM and LNO are right angles. Since right<br />
angles are congruent, LNM LNO. −−<br />
LN −−<br />
LN by <strong>the</strong> reflexive property of<br />
congruence. There<strong>for</strong>e, by ASA, LMN LON.<br />
1 b A correct conclusion is reached and reason is given, but no appropriate method<br />
is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–5 103
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
104 Cumulative Reviews<br />
Statements Reasons<br />
1. PET 1. Given.<br />
2. −−−<br />
PCB −−<br />
ET ; −−−<br />
TCA −−<br />
EP 2. Given.<br />
3. CBT and CAP are right<br />
angles.<br />
3. Definition of perpendicular lines.<br />
4. CBT CAP 4. Right angles are congruent.<br />
5. ACP BCP 5. Vertical angles are congruent.<br />
6. −−<br />
PA −−<br />
TB 6. Given.<br />
7. ACP BCT 7. AAS AAS.<br />
8. −−<br />
CP −−<br />
CT 8. CPCTC.<br />
9. PCT is isosceles. 9. Definition of isosceles triangle.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or<br />
three steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than three reasons are<br />
faulty or more than three steps are missing or have errors.<br />
1 PCT is isosceles and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
AC −−<br />
BD 1. Given.<br />
2. −−<br />
AC −−<br />
BC −−<br />
BD −−<br />
BC 2. Subtraction postulate.<br />
3. −−<br />
AB −−−<br />
CD 3. Partition postulate.<br />
4. −−<br />
GB −−−<br />
AD ; −−<br />
EC −−−<br />
AD 4. Given.<br />
5. GBA and ECD are right angles. 5. Definition of perpendicular lines.<br />
6. GBA ECD 6. Right angles are congruent.<br />
7. AGB DEC 7. Given.<br />
8. AGB DEC 8. AAS AAS.<br />
9. A D 9. CPCTC.<br />
10. −−<br />
AF −−<br />
DF 10. Converse of isosceles triangle <strong>the</strong>orem.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or<br />
three steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than three reasons are<br />
faulty or more than three steps are missing or have errors.<br />
1 −−<br />
AF −−<br />
DF and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–5 105
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
BD is <strong>the</strong> median of ABC. 1. Given.<br />
2. −−<br />
BA −−<br />
BC 2. Given.<br />
3. ABC is isosceles. 3. Definition of isosceles triangle.<br />
4. −−<br />
BD −−<br />
AC 4. The median from <strong>the</strong> vertex angle of an<br />
5.<br />
isosceles triangle is perpendicular to <strong>the</strong> base.<br />
−−<br />
BD is an altitude to −−<br />
AC . 5. Definition of an altitude.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty<br />
or multiple steps are missing or have errors.<br />
1 −−<br />
BD is an altitude to −−<br />
AC and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–6<br />
(pages 450–453)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) a ab b 2 6 (3) 4z<br />
2 (1) (x 9, y 5)<br />
3 (4)<br />
7 (2)<br />
−−<br />
AE −−<br />
BE 8 (2) −−<br />
XY −−−<br />
MA<br />
4 (2) (x, y) 9 (1) (4, 5)<br />
5 (4) ~r → ~p 10 (2) dilation<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 25, and an appropriate explanation is given.<br />
1 25, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
106 Cumulative Reviews
12 Score Explanation<br />
2 100, and an appropriate explanation is given.<br />
1 100, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 a (3, 4) b (3, 4) c (4, 3) d (3, 2)<br />
1 Student answers only two parts correctly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 39 and 51, and an appropriate explanation is given.<br />
1 39 and 51, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 45-45-90 isosceles triangle and an appropriate explanation is given.<br />
3 An appropriate method is used, but student does not identify <strong>the</strong> type of triangle.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 45-45-90 isosceles triangle, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 36, and an appropriate explanation is given.<br />
3 An appropriate method is used, but one computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 36, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–6 107
17 Score Explanation<br />
4 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
AB −−<br />
EB 1. Given.<br />
2. −−<br />
BC −−<br />
BD 2. Given.<br />
3. B B 3. Reflexive property of congruence.<br />
4. ABC EBD 4. SAS SAS.<br />
5. 1 2 5. CPCTC.<br />
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 1 2 and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
1 a K(1, 1), A(5, 2), T(3, 5)<br />
1 b K (1, 1) A(5, 2), T (3, 5)<br />
2 c<br />
y<br />
T'<br />
5<br />
4<br />
3<br />
T<br />
2<br />
A'<br />
1<br />
K' K<br />
5 4 3 2 1 1<br />
1<br />
K"<br />
2<br />
3<br />
4<br />
2 3 4<br />
A<br />
x<br />
5<br />
A"<br />
5<br />
T"<br />
2 d r x-axis<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
108 Cumulative Reviews
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. Isosceles ABC, with −−<br />
AB −−<br />
BC 1. Given.<br />
2. −−<br />
BD bisects ABC. 2. Given.<br />
3. ABD CBD 3. Definition of angle bisector.<br />
4. −−<br />
BD −−<br />
BD 4. Reflexive property of congruence.<br />
5. ABD CBD 5. SAS SAS.<br />
6. −−−<br />
AD −−−<br />
CD 6. CPCTC.<br />
7. −−<br />
BD bisects −−<br />
AC . 7. Definition of bisector.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 −−<br />
BD bisects −−<br />
AC and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
1 a C<br />
1 b D<br />
2 c B<br />
2 d C<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
Chapters 1–7<br />
(pages 454–456)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) supplementary 6 (4) −−<br />
AB −−−<br />
RM<br />
2 (4) (0, 6) 7 (2) 3(b a)<br />
3 (1) 6 8 (1) B is <strong>the</strong> largest angle.<br />
4 (3) I and III only 9 (1) 15<br />
5 (2) 2 10 (1) x _<br />
2<br />
Chapters 1–7 109
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 6, and an appropriate explanation is given.<br />
1 6, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 (7, 5), and an appropriate explanation is given.<br />
1 (7, 5), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 144, and an appropriate explanation is given.<br />
1 144, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 (3, 3), and an appropriate explanation is given.<br />
1 (3, 3), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
1 a (4, 0)<br />
b (4, 2)<br />
1 c (6, 3)<br />
d (3, 6)<br />
2 e (8, 3)<br />
f (6, 3)<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
110 Cumulative Reviews
16 Score Explanation<br />
4 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Right angles B and A are congruent because −−<br />
AB −−<br />
DB and<br />
−−<br />
AC −−−<br />
DC . −−−<br />
BW −−−<br />
CW . BWA CWD because vertical angles are congruent.<br />
There<strong>for</strong>e, by ASA ABW DCW. −−−<br />
AW −−−<br />
DW because of CPCTC. Hence, AWD is<br />
isosceles by definition of an isosceles triangle.<br />
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 AWD is isosceles and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. BC AB 1. Given.<br />
2. mBAC mBCA 2. The measures of <strong>the</strong> angles opposite unequal<br />
sides are unequal.<br />
3. 1 _ mBAC <br />
2 1 _ mBCA 3. Division postulate.<br />
2<br />
4. −−−<br />
DA bisects BAC. 4. Given.<br />
5. mDAC 1 _ mBAC 5. Definition of a bisector.<br />
2<br />
6. −−−<br />
DC bisects BCA. 6. Given.<br />
7. mDCA 1 _ mBCA 7. Definition of a bisector.<br />
2<br />
8. mDAC mDCA 8. Substitution postulate.<br />
9. DC DA 9. The lengths of <strong>the</strong> sides opposite two unequal<br />
angles of a triangle are unequal.<br />
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 DC DA and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–7 111
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
1 a (4, 1)<br />
1 b √ <br />
10<br />
2 c y 3x 13<br />
2<br />
d Slope −−−<br />
MD<br />
112 Cumulative Reviews<br />
<br />
1 _ , slope of <strong>the</strong> median is 3. The slopes are negative<br />
3<br />
reciprocals; <strong>the</strong>re<strong>for</strong>e, <strong>the</strong> median is perpendicular to −−−<br />
MD .<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
4 a Altitude from D to −−<br />
AC is x 0. Altitude from C to −−−<br />
AD is y 1 _ x 3.<br />
2<br />
Altitude from A to −−−<br />
CD is y x 3.<br />
3 a An appropriate method is used, but one computational error is made.<br />
2 a An appropriate method is used, but multiple computational errors are made.<br />
1 a The altitudes are identified, but no explanation is given.<br />
2 b (0, 3), and appropriate work is shown.<br />
1 b (0, 3), but no work is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. ABC, −−<br />
AB −−<br />
BC 1. Given.<br />
2. ABC is isosceles. 2. Definition of isosceles triangle.<br />
3. A C 3. Definition of isosceles triangle.<br />
4. mC mT 4. Definition of an exterior angle.<br />
5. mA mT 5. Substitution postulate.<br />
6. PT AP 6. Greater side opposite greater angle.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty<br />
or multiple steps are missing or have errors.
Chapters 1–8<br />
(pages 457–460)<br />
1 PT AP and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) rotation of 180 6 (1) y<br />
2 (1) 5 7 (3) (5, 3)<br />
3 (1) x 2y 5 8 (2) negative and less than <strong>the</strong> x-intercept of m<br />
4 (1) 23 9 (2) 2<br />
5 (1) {1, 2, 3} 10 (4) (9, 7)<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 m1 36, m2 72, m3 36, mAOE 144, and an appropriate explanation is<br />
given.<br />
1 m1 36, m2 72, m3 36, mAOE 144, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 (0, 1), and an appropriate explanation is given.<br />
1 (0, 1), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 (5, 1), and an appropriate explanation is given.<br />
1 (5, 1), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 105, and an appropriate explanation is given.<br />
1 105, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–8 113
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 2 : 3, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 2 : 3, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 x 7, mP 62, mQ 28, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 x 7, mP 62, mQ 28, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−−<br />
AD −−<br />
BC , −−<br />
DE −−<br />
CE , −−<br />
EA −−<br />
EB 1. Given.<br />
2. ADE BCE 2. SSS SSS.<br />
3. 1 2 3. CPCTC.<br />
4. 3 4 4. Base angles of an isosceles triangle are<br />
congruent.<br />
5. 1 3 2 4 5. Addition postulate.<br />
6. DAB CBA 6. Partition postulate.<br />
3 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 DAB CBA and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
114 Cumulative Reviews
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 2. If two sides are congruent, <strong>the</strong> triangle is isosceles.<br />
3. Base angles of isosceles triangles are congruent.<br />
4. Exterior angle of a triangle is greater than ei<strong>the</strong>r nonadjacent interior angle.<br />
5. Substitution postulate.<br />
6. Longest side of a triangle is opposite <strong>the</strong> angle with <strong>the</strong> largest measure.<br />
5 One reason is faulty or missing.<br />
4 Two reasons are faulty or missing.<br />
3 Three reasons are faulty or missing.<br />
2 More than three reasons are faulty or missing.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Check students’ graphs. The coordinates are listed below.<br />
19 Score Explanation<br />
1 a A(6, 6), B(12, 6), C(6, 15)<br />
2 b A(6, 6), B(12, 6), C(6, 15)<br />
2 c A(6, 4), B(6, 10), C(15, 4)<br />
1 d D 3<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
20 Score Explanation<br />
1 a Student graphs points A and B correctly.<br />
3 b x 4 and an appropriate explanation is given. Student graphs point C correctly.<br />
2 c 0, and an appropriate explanation is given.<br />
1 Correct answers <strong>for</strong> b and c, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–8 115
Chapters 1–9<br />
(pages 461–463)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) right<br />
2 (4) −−<br />
6 (1) 34<br />
AE bisects BAC.<br />
7 (3) SSS<br />
3 (4) 4, 5, 6<br />
8 (1) 290<br />
4 (1) (5, 3)<br />
9 (3) 3,600<br />
5 (2) congruent<br />
10 (1) 30<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 x is divisible by 2, and an appropriate explanation is given.<br />
1 x is divisible by 2, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 155, and an appropriate explanation is given.<br />
1 155, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 (9, 2), and an appropriate explanation is given.<br />
1 (9, 2), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 (4, 2), and an appropriate explanation is given.<br />
1 (4, 2), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
116 Cumulative Reviews
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 a True b True c False<br />
d True e False f True<br />
3 Student answers all but one part correctly.<br />
2 Student answers two or three parts incorrectly.<br />
1 Student answers four or five parts incorrectly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 18, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 18, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 a Yes b Yes c Yes<br />
d No e No f Yes<br />
3 Student answers all but one part correctly.<br />
2 Student answers two or three parts incorrectly.<br />
1 Student answers four or five parts incorrectly.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 AB √ 85 , BC √ 85 , AC √ 68 , and an appropriate explanation is given.<br />
5 An appropriate method is shown, but one computational mistake is made.<br />
4 An appropriate method is shown, but two computational mistakes are made.<br />
3 An appropriate method is shown, but wrong sides of <strong>the</strong> triangle are shown congruent.<br />
2 An appropriate method is shown, but more than two computational mistakes are<br />
made.<br />
1 An appropriate method is shown, but no conclusion is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–9 117
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
118 Cumulative Reviews<br />
Statements Reasons<br />
1. ABC is equilateral. 1. Given.<br />
2. mBAC 60 2. Interior angles of an equilateral<br />
triangle are 60.<br />
3. BCA and BCE are a linear pair. 3. Definition of linear pair.<br />
4. BCE is supplementary to BCA. 4. If two angles <strong>for</strong>m a linear pair, <strong>the</strong>n<br />
<strong>the</strong>y are supplementary.<br />
5. mBCA mBCE 180 5. Definition of supplementary angles.<br />
6. mBCE 120 6. Subtraction postulate.<br />
7. −−−<br />
CD bisects BCE. 7. Given.<br />
8. mDCE 1 _<br />
mBCE 8. Definition of angle bisector.<br />
2<br />
9. mDCE 60 9. Substitution.<br />
10. mDCE mBAC 10. Substitution.<br />
11. DCE BAC 11. Angles with equal measures are<br />
congruent.<br />
12. CD −−<br />
AB 12. Lines with congruent corresponding<br />
angles are parallel.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 CD −−<br />
AB and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
BC −−−<br />
AD 1. Given.<br />
2. −−<br />
BC −−−<br />
AD 2. Given.<br />
3. EAD FCB 3. Alternate interior angles are congruent.<br />
4. −−<br />
BF −−<br />
ED 4. Given.<br />
5. BFC DEA 5. Alternate interior angles are congruent.<br />
6. BFC DEA 6. AAS AAS.<br />
7. −−<br />
BF −−<br />
ED 7. CPCTC.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 −−<br />
BF −−<br />
ED and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–10<br />
(pages 464–466)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (2) 360<br />
2 (4) 140<br />
3 (1) 22<br />
4 (3) y x<br />
5 (3) 8<br />
6 (3) a rhombus<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
7 (1) y 1 _ x <br />
3 10 _<br />
3<br />
8 (4) The diagonals bisect <strong>the</strong> angles of<br />
<strong>the</strong> parallelogram.<br />
9 (1) adjacent<br />
10 (3) m4 mB<br />
2 120, and an appropriate explanation is given.<br />
1 120, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–10 119
12 Score Explanation<br />
2 31, and an appropriate explanation is given.<br />
1 31, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 7 √ 2 , and an appropriate explanation is given.<br />
1 7 √ 2 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 9, and an appropriate explanation is given.<br />
1 9, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 6, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 6, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 A(3, 6), B(2, 1), C(9, 1), and <strong>the</strong> trans<strong>for</strong>mations are per<strong>for</strong>med correctly.<br />
3 Student arrives at <strong>the</strong> correct answer, but per<strong>for</strong>ms one incorrect trans<strong>for</strong>mation.<br />
2 Student per<strong>for</strong>ms trans<strong>for</strong>mations in <strong>the</strong> wrong order.<br />
1 A(3, 6), B(2, 1), C(9, 1), but no appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
120 Cumulative Reviews
17 Score Explanation<br />
4 51, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 51, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
3 a Slope −−−<br />
DA<br />
<br />
1 _ , slope<br />
3 −− <br />
2<br />
AR _ , slope<br />
3 −− <br />
1<br />
RT _ ,<br />
3 −−<br />
TD is a vertical line. Since<br />
two sides have <strong>the</strong> same slope, −−−<br />
DA −−<br />
RT . DART is a trapezoid.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
3 b <strong>AK</strong> TA, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
3 a The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−−<br />
AD −−<br />
BE 1. Given.<br />
2. −−<br />
AE −−<br />
BD<br />
3.<br />
2. Given.<br />
___<br />
DE ___<br />
DE 3. Reflexive property of congruence.<br />
4. ADE BED 4. SSS SSS.<br />
5. BDE AED 5. CPCTC.<br />
1 BDE AED and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
3 b The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−−<br />
AD −−<br />
BE 1. Given.<br />
2. −−<br />
AE −−<br />
BD 2. Given.<br />
3. −−<br />
AB −−<br />
AB 3. Reflexive property of congruence.<br />
4. ADB BEA 4. SSS SSS.<br />
5. FAB FBA 5. CPCTC.<br />
1 FAB FBA and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–10 121
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
122 Cumulative Reviews<br />
Statements Reasons<br />
1. Quadrilateral ABCD and<br />
diagonal −−−−<br />
AFEC<br />
1. Given.<br />
2. −−<br />
BF −−<br />
AC , −−<br />
DE −−<br />
AC 2. Given.<br />
3. BFA and DEC are right<br />
angles.<br />
3. Definition of perpendicular lines.<br />
4. BFA DEC 4. Right angles are congruent.<br />
5. −−<br />
BF −−<br />
DE , −−<br />
AE −−<br />
CF 5. Given.<br />
6. ADE CBF 6. SAS SAS.<br />
7. −−−<br />
AD −−<br />
BC 7. CPCTC.<br />
8. DAE BCF 8. CPCTC.<br />
9. −−−<br />
AD −−<br />
BC 9. Alternate interior angles of two parallel lines<br />
are congruent.<br />
10. ABCD is a parallelogram. 10. Definition of a parallelogram.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 ABCD is a parallelogram and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–11<br />
(pages 467–469)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (1) b _<br />
a<br />
6 (3) 160 cm<br />
2 (3) equilateral and equiangular<br />
2<br />
7 (1) 2 √ 6 <br />
3 (3) 120<br />
4 (1) −−<br />
AC<br />
5 (2) 9<br />
8 (3) 3 _<br />
5<br />
9 (3) The sphere is inscribed in <strong>the</strong> cube.<br />
10 (1) parallelogram
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 x 116, y 32, and an appropriate explanation is given.<br />
1 x 116, y 32, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 n 1, and an appropriate explanation is given.<br />
1 n 1, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 116.5, and an appropriate explanation is given.<br />
1 116.5, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 50, and an appropriate explanation is given.<br />
1 50, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 3 √ 10 , and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used, but multiple arithmetical errors are made.<br />
1 3 √ 10 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–11 123
16 Score Explanation<br />
4 DN √ 106 , NA √ 106 , making DNA isosceles.<br />
Slope −−−<br />
9<br />
DN _ and slope<br />
5 −−−<br />
5<br />
NA _ . Since <strong>the</strong> slopes are negative reciprocals of each<br />
9<br />
o<strong>the</strong>r, −−−<br />
DN −−−<br />
NA . There<strong>for</strong>e, DNA is an isosceles right triangle.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but DNA is not proven to be an isosceles right<br />
triangle.<br />
1 An appropriate method is used, but multiple computational errors are made.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 6 √ , and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 6 √ , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
124 Cumulative Reviews
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−−<br />
DA −−<br />
AB , −−−<br />
DC −−<br />
CB 1. Given.<br />
2. BAD and BCD are right angles. 2. Definition of perpendicular lines.<br />
3. BAD BCD 3. Right angles are congruent.<br />
4. 1 2 4. Given.<br />
5. BAE BCE 5. Subtraction postulate.<br />
6. −−<br />
BD −−<br />
AC at E 6. Given.<br />
7. BEA and BEC are right angles. 7. Definition of perpendicular lines.<br />
8. BEA BEC 8. Right angles are congruent.<br />
9. −−<br />
BE −−<br />
BE 9. Reflexive property of congruence.<br />
10. ABE CBE 10. AAS AAS.<br />
11. 3 4 11. CPCTC.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/<br />
or one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 3 4 and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
2 a Student proves that MIKE is a rhombus by showing that<br />
MI IK KE EM √ 40 or by some o<strong>the</strong>r appropriate method.<br />
1 a The correct conclusion is reached and a reason is given, but no appropriate method<br />
is shown.<br />
2 b Student shows that MIKE is not a square by showing that <strong>the</strong><br />
diagonals are not perpendicular. Slope −− <br />
1<br />
MI _ and <strong>the</strong> slope<br />
3 −− 3.<br />
IK<br />
1 b The correct conclusion is reached and a reason is given, but no appropriate method<br />
is shown.<br />
2 c 32, and an appropriate explanation is given.<br />
1 c 32, but no appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–11 125
20 Score Explanation<br />
2 a 3, and an appropriate explanation is given.<br />
1 3, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
2 b 36, and an appropriate explanation is given.<br />
1 36, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
2 c 1 _ , and an appropriate explanation is given.<br />
4<br />
1 1 _ , but no explanation is given.<br />
4<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–12<br />
(pages 470–472)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) It has a slope of 0.<br />
2 (2) are congruent<br />
3 (3) 50<br />
4 (3) cylinder<br />
5 (4) 130<br />
6 (3) 16 x 2<br />
7 (2) 10 √ 2<br />
8 (1) 1 : 3<br />
9 (1) 4 _<br />
25<br />
10 (3) 4x y 2<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 225, and an appropriate explanation is given.<br />
1 225, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
126 Cumulative Reviews
12 Score Explanation<br />
2 (0, 4) and (6, 0), and an appropriate explanation is given.<br />
1 (0, 4) and (6, 0), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
1 a 9 _ , and an appropriate explanation is given.<br />
16<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
1 b 27 _ , and an appropriate explanation is given.<br />
64<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 102, and an appropriate explanation is given.<br />
1 102, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
2 a 22, and an appropriate explanation is given.<br />
1 a 22, but no appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
2 b 78, and an appropriate explanation is given.<br />
1 b 78, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 2 √ 3 , and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 2 √ 3 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–12 127
17 Score Explanation<br />
4 6 √ <br />
5 , and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 6 √ 5 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 Student uses an appropriate method, such as showing that both pairs of opposite sides<br />
are parallel, by showing that slope −−− 0, slope −−− 0; slope −−− <br />
b<br />
AD ME MA _<br />
a<br />
, slope −− <br />
b<br />
ED _<br />
a<br />
.<br />
5 An appropriate method is used, but one computational error is made.<br />
4 An appropriate method is used, but two computational errors are made.<br />
3 An appropriate method is used, but three computational errors are made.<br />
2 More than three computational errors are made.<br />
1 An appropriate method is used, but does not prove that MADE is a parallelogram.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
128 Cumulative Reviews
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. ABC is isosceles. 1. Given.<br />
2. −−<br />
AB −−<br />
AC 2. Definition of isosceles triangle.<br />
3. AMC is isosceles. 3. Given.<br />
4. −−−<br />
AM −−−<br />
CM 4. Definition of isosceles triangle.<br />
5. −−−<br />
PM −−−<br />
QM 5. Given.<br />
6. −−−<br />
AM −−−<br />
QM −−−<br />
CM −−−<br />
PM 6. Addition postulate.<br />
7. −−−<br />
AQ −−<br />
CP 7. Partition postulate.<br />
8. BAC BCA 8. Isosceles triangle <strong>the</strong>orem.<br />
9. MAC MCA 9. Isosceles triangle <strong>the</strong>orem.<br />
10. BAC MAC <br />
BCA MCA<br />
10. Subtraction postulate.<br />
11. BAM BCP 11. Partition postulate.<br />
12. ABQ CBP 12. SAS SAS.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 ABQ CBP and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–12 129
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
130 Cumulative Reviews<br />
Statements Reasons<br />
1. Parallelogram ABCD 1. Given.<br />
2. A C 2. Opposite angles of a parallelogram are<br />
congruent.<br />
3. m1 m2 3. Given.<br />
4. 1 2 4. Angles with equal measures are<br />
congruent.<br />
5. 1 and 4 are a linear pair.<br />
2 and 3 are a linear pair.<br />
6. 1 and 4 are supplementary. 2 and<br />
3 are supplementary.<br />
5. Definition of linear pair.<br />
6. Two angles that <strong>for</strong>m a linear pair are<br />
supplementary.<br />
7. 3 4 7. Supplements of congruent angles are<br />
congruent.<br />
8. AFE CHG 8. AA.<br />
9. FE _<br />
FA<br />
GH<br />
_<br />
CH<br />
9. Corresponding parts of similar<br />
triangles are in proportion.<br />
10. FE CH GH FA 10. The product of means equals <strong>the</strong><br />
product of extremes.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 FE CH GH FA and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–13<br />
(pages 473–475)<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (4) ABD CDB<br />
2 (2) a median<br />
3 (1) 12<br />
4 (1) 1 : 2<br />
5 (1) 2<br />
6 (3) R 270<br />
7 (1) 30<br />
8 (1) 12<br />
9 (3) III and IV<br />
10 (4) −−<br />
TA −−<br />
TO
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 (16, 14), and an appropriate explanation is given.<br />
1 (16, 14), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 55, and an appropriate explanation is given.<br />
1 55, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
13 Score Explanation<br />
2 38, and an appropriate explanation is given.<br />
1 38, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 30.5, and an appropriate explanation is given.<br />
1 30.5, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 y x 3, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but <strong>the</strong> equation of a parallel line is given.<br />
1 y x 3, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–13 131
16 Score Explanation<br />
4 mADE 93, mBGA 87, mABC 87, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but one of <strong>the</strong> angle measures found is incorrect.<br />
1 mADE 93, mBGA 87, mABC 87, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 h r _<br />
, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but h is found in terms of r and s.<br />
1 h r _<br />
, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
6 h 12, and an appropriate explanation is given.<br />
5 An appropriate method is used, but one computational error is made.<br />
4 An appropriate method is used, but two computational errors are made.<br />
3 An appropriate method is used, but three computational errors are made.<br />
2 More than three computational errors are made.<br />
1 h 12, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
132 Cumulative Reviews
19 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. Parallelogram ABCD;<br />
−−−<br />
EAB ; −−−<br />
DCF<br />
1. Given.<br />
2. −−<br />
AB −−−<br />
DC 2. Definition of a parallelogram.<br />
3. −−<br />
EB −−<br />
DF 3. Collinearity.<br />
4. AEH CFG 4. Alternate interior angles are congruent.<br />
5. BAH BCD 5. Opposite angles of a parallelogram are<br />
congruent.<br />
6. GCF and BCD are a linear<br />
pair. HAE and BAH are a<br />
linear pair.<br />
7. GCF and BCD are supplementary.<br />
HAE and BAH are<br />
supplementary.<br />
6. Definition of linear pair.<br />
7. Two angles that <strong>for</strong>m a linear pair are<br />
supplementary.<br />
8. GCF HAE 8. Supplements of congruent angles are<br />
congruent.<br />
9. a EAH FCG 9. AA.<br />
10. EA _<br />
AH<br />
FC<br />
_<br />
CG<br />
10. Corresponding parts of similar triangles are in<br />
proportion.<br />
11. b EA CG AH CG 11. The product of means equals <strong>the</strong> product of<br />
extremes.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 An appropriate proof with correct conclusion is shown <strong>for</strong> part a only.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than two reasons are<br />
faulty or more than two steps are missing or have errors.<br />
1 EAH FCG and EA CG AH CG, and a reason is given, but no appropriate<br />
method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–13 133
20 Score Explanation<br />
Chapters 1–14<br />
(pages 476–479)<br />
1 a m <br />
BC 72, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
1 b mEDC 108, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
1 c mBCR 36, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
1 d m<strong>AK</strong>B 72, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
2 e mRF 72, and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part I<br />
Allow a total of 20 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (1) x 3<br />
2 (3) 3 : 5<br />
3 (3) 125<br />
4 (2) 45<br />
5 (3) 2 √ 2<br />
134 Cumulative Reviews<br />
6 (2) 8.8<br />
7 (2) 2x 3<br />
8 (1) {C, D, G, H}<br />
9 (4) y 3x 5<br />
10 (4) similar triangles<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
11 Score Explanation<br />
2 720, and an appropriate explanation is given.<br />
1 720, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
12 Score Explanation<br />
2 144, and an appropriate explanation is given.<br />
1 144, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
13 Score Explanation<br />
2 13, and an appropriate explanation is given.<br />
1 13, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
14 Score Explanation<br />
2 80, and an appropriate explanation is given.<br />
1 80, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
15 Score Explanation<br />
4 Each side is 2b, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but h is found in terms of r and s.<br />
1 Each side is 2b, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
16 Score Explanation<br />
4 3 _ , and an appropriate explanation is given.<br />
10<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but h is found in terms of r and s.<br />
1 3 _ , but no explanation is given.<br />
10<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
17 Score Explanation<br />
4 30,776.3 in. 3 , and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but h is found in terms of r and s.<br />
1 30,776.3 in. 3 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–14 135
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
18 Score Explanation<br />
3 a<br />
x y 7<br />
136 Cumulative Reviews<br />
y<br />
10<br />
9<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
3y 2x 6<br />
7 6 5 4 3 2 1<br />
1<br />
1 2 3 4 5 6 7 8 9 10<br />
y 2<br />
2<br />
3<br />
4<br />
5<br />
6<br />
7<br />
x<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
3 b 45, and an appropriate explanation is given.<br />
2 An appropriate method is used, but a single computational error is made.<br />
1 45, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
19 Score Explanation<br />
4 a Slope −−−<br />
AX<br />
1 and slope −−− 1. Since <strong>the</strong> slopes are negative reciprocals of each<br />
ME<br />
o<strong>the</strong>r, <strong>the</strong> lines are perpendicular. The midpoint of −−−<br />
AX is<br />
(4, 3), which is point E. There<strong>for</strong>e, −−−<br />
ME is <strong>the</strong> perpendicular bisector.<br />
3 An appropriate method is used, but a single arithmetical error is made.<br />
2 An appropriate method is used to show <strong>the</strong> lines are perpendicular, but no midpoint is<br />
found.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
2 b Shows that MAX is isosceles by MX MA √ 68 , and an appropriate explanation<br />
is given.<br />
1 An appropriate method is used, but a single computational error is made.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
20 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. AE CE 1. Given.<br />
2. ___<br />
AE ___<br />
CE 2. Congruent arcs have congruent chords.<br />
3. ABE EBD 3. Congruent central angles have congruent<br />
chords.<br />
4. −−<br />
FE is tangent to O at E. 4. Given.<br />
5. FEB is a right angle. 5. Definition of a tangent.<br />
6. FEB and BED are a linear<br />
pair.<br />
7. FEB and BED are<br />
supplementary.<br />
6. Definition of linear pair.<br />
7. Two angles that <strong>for</strong>m a linear pair are<br />
supplementary.<br />
8. BED is a right angle. 8. A supplement to a right angle is a right<br />
angle.<br />
9. Diameter −−−<br />
BOE 9. Given.<br />
10. <br />
BCE is a semicircle. 10. Definition of a semicircle.<br />
11. BAE is a right angle. 11. An angle inscribed in a semicircle is always a<br />
right angle.<br />
12. BAE BED 12. Right angles are congruent.<br />
13. ABE EBD 13. AA.<br />
14. BE _<br />
BA<br />
<br />
BD _<br />
BE<br />
14. Corresponding parts of similar triangles are in<br />
proportion.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but two reasons are faulty or<br />
two steps are missing.<br />
2 An appropriate proof with correct conclusion is shown, but multiple reasons are faulty<br />
or multiple steps are missing or have errors.<br />
1 BE _ <br />
BD<br />
BA _ , and a reason is given, but no appropriate method is shown.<br />
BE<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Chapters 1–14 137
Practice <strong>Regents</strong> <strong>Examination</strong> One<br />
(pages 483–489)<br />
<br />
Part I<br />
Allow a total of 56 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (3) 1 _<br />
2<br />
2 (2) 8 √ 2<br />
3 (1) 1 _ b<br />
2 2<br />
4 (1) 48<br />
5 (2) 8 √ 3<br />
6 (2) 12<br />
7 (4) r : 2<br />
8 (4) 106<br />
9 (1) 8<br />
10 (3) AEC is a<br />
right angle.<br />
138<br />
11 (3) (9, 3)<br />
12 (1) 6<br />
13 (4) 16 : 81<br />
14 (3) 18<br />
15 (4) 1<br />
16 (2) 26<br />
17 (4) 16<br />
18 (2) y<br />
19 (2) 6 √ 2 <br />
20 (1) 86<br />
21 (2) 18<br />
22 (2) (0, 1)<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
29 Score Explanation<br />
Practice <strong>Geometry</strong><br />
<strong>Regents</strong> <strong>Examination</strong>s<br />
23 (4) 4<br />
24 (3) right<br />
25 (2) 9<br />
26 (3) x 3<br />
27 (4) I and II only<br />
28 (3) 1 _ V<br />
8<br />
2 (x 2) 2 (y 4) 2 2.25, and an appropriate explanation is given.<br />
1 (x 2) 2 (y 4) 2 2.25, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.
30 Score Explanation<br />
2 100, and an appropriate explanation is given.<br />
1 100, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
31 Score Explanation<br />
2 30, and an appropriate explanation is given.<br />
1 30, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
32 Score Explanation<br />
2 3 √ 2 , and an appropriate explanation is given.<br />
1 3 √ 2 , but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
33 Score Explanation<br />
2 16, and an appropriate explanation is given.<br />
1 16, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
34 Score Explanation<br />
2 (1, 0), and an appropriate explanation is given.<br />
1 (1, 0), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
35 Score Explanation<br />
4 y x 1, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 y x 1, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Practice <strong>Regents</strong> <strong>Examination</strong> One 139
36 Score Explanation<br />
4 36, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 36, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
37 Score Explanation<br />
4 27, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 27, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
38 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. −−<br />
LA −−<br />
EG 1. Given.<br />
2. LAR GER 2. Alternate interior angles are congruent.<br />
3. ALR EGR 3. Alternate interior angles are congruent.<br />
4. ALR EGR 4. AA.<br />
5. AR _ <br />
RE<br />
RL _<br />
RG<br />
5. Corresponding parts of similar triangles are in proportion.<br />
6. Rhombus PARL 6. Given.<br />
7. AR PA; RL PL 7. Definition of a rhombus.<br />
8. AR RG RL RE 8. The product of means equals <strong>the</strong> product of extremes.<br />
9. PA RG RE PL 9. Substitution postulate.<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from <strong>the</strong> statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but three reasons are faulty or<br />
three steps are missing.<br />
2 An appropriate proof with correct conclusion is used, but more than three reasons are<br />
faulty or more than three steps are missing or have errors.<br />
1 PA RG RE PL and a reason is given, but no appropriate method is shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
140 Practice <strong>Geometry</strong> <strong>Regents</strong> <strong>Examination</strong>s
Practice <strong>Regents</strong> <strong>Examination</strong> Two<br />
(pages 490–496)<br />
Part I<br />
Allow a total of 56 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (1) −−<br />
EK −−<br />
EY<br />
2 (4) 150<br />
3 (1) 4 √ 3<br />
4 (3) 1 _<br />
2<br />
5 (3) 36<br />
6 (4) R 90<br />
7 (4) Use <strong>the</strong> slope <strong>for</strong>mula<br />
to prove diagonals are<br />
perpendicular.<br />
8 (2) 105<br />
9 (4) {2, 5, √ 29 }<br />
10 (2) 120<br />
11 (2) 20 √ 3 <br />
12 (3) 24<br />
13 (1) 28<br />
14 (3) 160<br />
15 (3) 2<br />
16 (1) 4 _ 3<br />
units<br />
3<br />
17 (4) 6 s 2x<br />
18 (4) 11<br />
19 (3) 8<br />
20 (3) 15<br />
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
29 Score Explanation<br />
21 (2) 5<br />
22 (1) 64 cm 3<br />
23 (4) 4<br />
24 (3) perpendicular<br />
bisectors of <strong>the</strong> sides<br />
of <strong>the</strong> triangle<br />
25 (2) 5 √ 2 <br />
26 (4) 36<br />
27 (1) BC AB<br />
28 (3) equal in area<br />
2 69, and an appropriate explanation is given.<br />
1 69, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
30 Score Explanation<br />
2 (x 4) 2 (y 5) 2 9, and an appropriate explanation is given.<br />
1 (x 4) 2 (y 5) 2 9, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
31 Score Explanation<br />
2 (2x, 3y), and an appropriate explanation is given.<br />
1 (2x, 3y), but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Practice <strong>Regents</strong> <strong>Examination</strong> Two 141
32 Score Explanation<br />
2 108, and an appropriate explanation is given.<br />
1 108, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
33 Score Explanation<br />
2 6.8, and an appropriate explanation is given.<br />
1 6.8, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
34 Score Explanation<br />
2 117, and an appropriate explanation is given.<br />
1 117, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct<br />
response that was obtained by an obviously incorrect procedure.<br />
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
35 Score Explanation<br />
4 11.55, and an appropriate explanation is given.<br />
3 An appropriate method is used, but answer is not rounded to <strong>the</strong> nearest hundredth.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 11.55, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
36 Score Explanation<br />
4 OC 21, mCOD 69.5, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but only one of <strong>the</strong> two answers is found.<br />
1 OC 21, mCOD 69.5, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
142 Practice <strong>Geometry</strong> <strong>Regents</strong> <strong>Examination</strong>s
37 Score Explanation<br />
4 Secant points of intersection are (0, 5) and (2, 4), and an appropriate explanation is<br />
given.<br />
3 Secant is given, but graph is incorrect.<br />
2 Graph is shown, but secant is not given.<br />
1 Secant is given, but graph is not shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
38 Score Explanation<br />
6 Any appropriate method is used to prove that ABCD is a parallelogram, but not a rectangle.<br />
For example: showing that <strong>the</strong> diagonals bisect each o<strong>the</strong>r but are not congruent,<br />
or showing that no consecutive sides have negative reciprocal slopes.<br />
5 An appropriate method is used, but a single computational error is made.<br />
4 An appropriate method is used, but two computational errors are made.<br />
3 An appropriate method is used to prove that ABCD is a parallelogram but does not<br />
show that it is not a rectangle.<br />
2 An appropriate method is used to prove that ABCD is a parallelogram, but more than<br />
two computational errors are made and student does not show that it is not a rectangle.<br />
1 An appropriate method is used, but ABCD is not shown to be a parallelogram, and<br />
multiple computational errors are made.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Practice <strong>Regents</strong> <strong>Examination</strong> Three<br />
(pages 497–504)<br />
Part 1<br />
Allow a total of 56 credits, 2 credits <strong>for</strong> each of <strong>the</strong> following correct answers. Allow credit if <strong>the</strong> student has<br />
written <strong>the</strong> correct answer instead of <strong>the</strong> numeral 1, 2, 3, or 4.<br />
1 (2) 50<br />
2 (4) 16 : 25<br />
3 (4) (7, 7 √ 3 )<br />
4 (2) 18 cm 2<br />
5 (3) {2, 7, 10}<br />
6 (3) (3, 6.5)<br />
7 (3) 80 √ 3<br />
8 (3) ABCD is a parallelogram<br />
whose diagonals<br />
bisect each o<strong>the</strong>r.<br />
9 (2) 2 _ <br />
3<br />
10 (2) 156<br />
11 (2) (x 3) 2 (y 7) 2 2<br />
12 (3) y 1 _ x 4<br />
2<br />
13 (3) d 5<br />
14 (4) 11<br />
15 (1) 60<br />
16 (2) 32 √ 3<br />
17 (2) 16 16 √ 2 <br />
18 (4) x 1<br />
19 (1) 5<br />
20 (2) y 5x 2<br />
21 (3) 4 √ 3<br />
22 (1) perpendicular bisectors<br />
of <strong>the</strong> sides<br />
23 (3) 120<br />
24 (4) obtuse<br />
25 (3) 60<br />
26 (4) 210<br />
27 (1) The perpendicular<br />
bisector of <strong>the</strong> chord<br />
of a circle bisects <strong>the</strong><br />
intercepted arc.<br />
28 (1) (3, 5)<br />
Practice <strong>Regents</strong> <strong>Examination</strong> Three 143
Part II<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 2 credits.<br />
29 Score Explanation<br />
2 216 cubic inches, and an appropriate explanation is given.<br />
1 216 cubic inches, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
30 Score Explanation<br />
1 a PA 9, and an appropriate explanation is given.<br />
1 b m CB 53 and an appropriate explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
31 Score Explanation<br />
2 16 inches, and an appropriate explanation is given.<br />
1 16 inches, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
32 Score Explanation<br />
2 60 _ , and an appropriate explanation is given.<br />
13<br />
1 60 _ , but no explanation is given.<br />
13<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
33 Score Explanation<br />
2 32, and an appropriate explanation is given.<br />
1 32, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
34 Score Explanation<br />
2 4, and an appropriate explanation is given.<br />
1 4, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
144 Practice <strong>Geometry</strong> <strong>Regents</strong> <strong>Examination</strong>s
Part III<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 4 credits.<br />
35 Score Explanation<br />
4 16, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but incorrect radius is found.<br />
1 16, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
36 Score Explanation<br />
4 40, and an appropriate explanation is given.<br />
3 An appropriate method is used, but a single computational error is made.<br />
2 An appropriate method is used, but multiple computational errors are made.<br />
1 40, but no explanation is given.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
37 Score Explanation<br />
4 2. Reflexive property of congruence<br />
3. Perpendicular lines meet to <strong>for</strong>m right angles.<br />
4. All right angles are congruent.<br />
5. Two right triangles are similar if an acute angle of one triangle is congruent to an<br />
acute angle of <strong>the</strong> o<strong>the</strong>r.<br />
6. Corresponding sides of similar triangles are in proportion.<br />
7. The product of means equals <strong>the</strong> product of extremes.<br />
3 One reason is incorrect.<br />
2 Two reasons are incorrect.<br />
1 More than two reasons are incorrect.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.<br />
Practice <strong>Regents</strong> <strong>Examination</strong> Three 145
Part IV<br />
For each question, use <strong>the</strong> specific criteria to award a maximum of 6 credits.<br />
38 Score Explanation<br />
6 The following proof or ano<strong>the</strong>r appropriate proof is given.<br />
Statements Reasons<br />
1. Parallelogram ABCD,<br />
with −−−<br />
ABG<br />
146 Practice <strong>Geometry</strong> <strong>Regents</strong> <strong>Examination</strong>s<br />
1. Given.<br />
2. −−<br />
AB −−−<br />
CD , −−−<br />
AG −−−<br />
CD 2. Definition of a parallelogram.<br />
3. MGB MDC 3. Alternate interior angles are congruent.<br />
4. BMG CMD 4. Vertical angles are congruent.<br />
5. M is <strong>the</strong> midpoint of −−<br />
BC . 5. Given.<br />
6. −−−<br />
BM −−−<br />
MC 6. Definition of midpoint.<br />
7. a BGM CDM 7. AAS AAS.<br />
8. −−<br />
BG −−−<br />
CD 8. CPCTC.<br />
9. −−<br />
AB −−−<br />
CD 9. Opposite sides of a parallelogram are congruent.<br />
10. b −−<br />
AB −−<br />
BG 10. Transitive postulate of congruence<br />
5 An appropriate proof with correct conclusion is shown, but one reason is faulty and/or<br />
one statement is missing.<br />
4 A faulty conclusion or no conclusion is drawn from correct statements and reasoning.<br />
3 An appropriate proof with correct conclusion is shown, but no conclusion is made<br />
<strong>for</strong> −−<br />
AB −−<br />
BG .<br />
2 An appropriate proof with correct conclusion is used, but multiple reasons are faulty or<br />
multiple steps are missing or have errors.<br />
1 BGM CDM and −−<br />
AB −−<br />
BG , and a reason is given, but no appropriate proof is<br />
shown.<br />
0 A zero response is completely incorrect, irrelevant, or incoherent or is a correct response<br />
that was obtained by an obviously incorrect procedure.