Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK Preparing for the Regents Examination Geometry, AK
3-4 Indirect Proof (page 45) 1 Statements 1. Reasons −−− CD and −−− HK are not congruent. 1. Assumption. 2. CD HK 3. 2. Given. −−− CD −−− HK 3. Line segments that are equal in measure are congruent. Therefore, assumption is false. 2 Statements Reasons 1. ABC is not a 1. Assumption. right angle. 2. −− AB −−− CD 2. Given. 3. ABC is a right 3. Perpendicular angle. lines form right angles. Therefore, the assumption is false. 3 Statements Reasons 1. ABC is not an isosceles triangle. 1. Assumption. 2. A B 2. Given. 3. ABC is an isos- 3. An isosceles trianceles triangle. gle contains two congruent angles. Therefore, the assumption is false. 4 Statements 1. DB is the angle bisector of ADC. Reasons 1. Assumption. 2. mADB mBDC 2. Given. 3. DB is not the bi- 3. An angle bisector sector of ADC. divides an angle into two congruent parts. Therefore, the assumption is false. 3-5 Postulates, Theorems, and Proof (pages 47–49) 1 Yes 2 No 3 No 4 The symmetric property of equality 5 The reflexive property of congruence 6 The symmetric property and transitive property of congruence 7 Statements Reasons 1. −− PQ −−− QR 1. Given. 2. PQR is a right angle. 2. Perpendicular lines are two lines that intersect to form right angles. 3. mPQR 90 3. A right angle is an angle whose degree measure is 90. 4. −− XY −− YZ 4. Given. 5. XYZ is a right angle. 5. Perpendicular lines are two lines that intersect to form right angles. 6. mXYZ 90 6. A right angle is an angle whose degree measure is 90. 7. 90 mXYZ 7. Symmetric property of equality. 8. mPQR mXYZ 8. Transitive property of equality. 8 Statements Reasons 1. AC is the angle 1. Given. bisector of BAD. 2. BAC CAD 2. An angle bisector divides an angle into two congruent parts. 3. mBAC mCAD 4. AD is the angle bisector of CAE. 3. If two angles are congruent, they have the same measure. 4. Given. 5. CAD DAE 5. An angle bisector divides an angle into two congruent parts. 3-5 Postulates, Theorems, and Proof 9
6. BAC DAE 6. Transitive property. 7. mBAC mDAE 7. If two angles are congruent, their measures are equal. 9 Statements Reasons 1. 8 x y 1. Given. 2. y 3 2. Given. 3. 8 x 3 3. Substitution property. 4. x 3 8 4. Symmetric property. 10 Statements Reasons 1. M is the midpoint of −− AB . 1. Given. 2. −−− AM −−− MB 2. A midpoint divides a line segment into two congruent line segments. 3. −−− MB −− BC 3. Given. 4. −−− AM −− BC 4. Transitive property. 3-6 Remaining Postulates of Equality (pages 51–52) 1 Partition postulate of equality 2 Division postulate of equality 3 Addition postulate 4 Subtraction postulate 5 Division postulate of equality Note: Since there are many variations of proofs, the following is simply one set of acceptable statements to complete each proof. Depending on the textbook used, the wording and format of reasons may differ, so they have not been supplied for the method of congruence applied in each problem. (These solutions are intended to be used as a guide—other possible solutions may vary.) 10 Chapter 3: Introduction to Geometric Proof 6 1. m1 m2 2. m3 m4 3. m1 m3 m2 m4 4. mDAB m1 m3 mBCD m2 m4 5. mDAB mBCD (Substitution 7 1. postulate) −− AB −− CB 2. −−− AD −− CE 3. AB CB 4. AD CE 5. AB AD CB CE 6. DB EB 7. −− DB −− EB 8 1. AB AC 2. AD (Line segments that are equal in measure are congruent.) 1 _ AC 3 3. AE 1 _ AB 3 4. AD AE (Division postulate) 9 1. mEAB mFBC 2. AG is the angle bisector of EAB. 3. BH is the angle bisector of FBC. 4. m1 1 _ mEAB 2 5. m2 1 _ mFBC 2 6. m1 m2 10 1. AB DE 2. AC 3AB 3. DF 3DE (Division postulate) 4. AC DF (Multiplication postulate) Chapter Review (page 52) 1 Reflexive property of equality 2 Transitive property of equality 3 Symmetric property of equality 4 mBAD 1 _ mBAC 2
- Page 1 and 2: ANSWER KEY Preparing for the REGENT
- Page 3 and 4: Contents Chapter 1: Essentials of G
- Page 5 and 6: 1-4 Angles (pages 9-10) 1 (4) It is
- Page 7 and 8: 6 True 7 True 8 True 9 True 10 Answ
- Page 9 and 10: 3 1. e ∨ ~f 2. ~f → g 3. ~e 4.
- Page 11: CHAPTER 3 3-1 Inductive Reasoning (
- Page 15 and 16: CHAPTER 4-1 Setting Up a Valid Proo
- Page 17 and 18: 6 Given: BA bisects CBE 1 2 2 4
- Page 19 and 20: 1 1. −−−− ABCD 2. 1 2 3. _
- Page 21 and 22: 18 1. I is the midpoint of −− E
- Page 23 and 24: 5-3 Isosceles and Equilateral Trian
- Page 25 and 26: 12. −−− HG −−− DC 13.
- Page 27 and 28: 5 1. −− FG is the perpendicular
- Page 29 and 30: Chapter Review (pages 84-85) Note:
- Page 31 and 32: 20 Use constructing congruent angle
- Page 33 and 34: 6-3 Line Reflections and Symmetry (
- Page 35 and 36: 11 A(0, 8), B(2, 2), C(6, 4) (8) 10
- Page 37 and 38: 5 (3) (x, y) → (x, 2y) 6 (1) tran
- Page 39 and 40: 11. mABC mADC 12. 2mABD 2mADB 13.
- Page 41 and 42: 6. mDAB mCAD mDCB mACD 7. mCAB
- Page 43 and 44: g e 17 a _ f d b Undefined c a _
- Page 45 and 46: 5 BIG is isosceles because it has t
- Page 47 and 48: ___ 27 a M KA (5, 1), M ___ AT (4
- Page 49 and 50: 8 106 9 mA 75, mC 67 10 57 11 60
- Page 51 and 52: 9-5 The Sum of the Measures of the
- Page 53 and 54: 9-7 The Converse of the Isosceles T
- Page 55 and 56: 12 1. ___ CE ___ BA , ___ BD ___
- Page 57 and 58: 12 a mx 45, my 45 b mx 98, my 8
- Page 59 and 60: Quadrilaterals 10-2 The Parallelogr
- Page 61 and 62: 7. MAD RCB 8. MAD RCB (SAS SAS)
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
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