Preparing for the Regents Examination Geometry, AK

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23 Converse: If a number is real, then it is rational. (False) Conditional statement is true. 24 Converse: If x 4 √ 2 , then x √ 32 . (True) Conditional statement is true. Biconditional: x √ 32 if and only if x 4 √ 2 . 25 Converse: If a relation is a function, then it passes the vertical line test. (True) Conditional statement is true. Biconditional: A relation passes the vertical line test if and only if it is a function. 26 If I study, then I will not get poor grades. 27 If I get poor grades, then I did not study. 28 If I do not get poor grades, then I study. 29 If my baby cousin cries, then she gets a bottle. 30 If the figure is a rectangle, then two adjacent sides form a right angle. 31 If a triangle is equilateral, then it is equiangular. 32 I like math. 33 The class president is a girl. (Law of Disjunctive Inference) 34 My teacher will be happy. (Law of Detachment) 35 I do not go to the park. (Law of Modus Tollens) 36 If I learn to knit, I will save money. (Chain Rule) 37 I eat lunch. (Law of Modus Tollens) 38 I feel good. (Law of Detachment) 39 Rich likes lacrosse. (Law of Disjunctive Inference) 40 Marlene does not babysit. (Law of Modus Tollens) 41 Jack is using his computer. (Law of Modus Tollens) 42 If I read novels when my teacher assigns them, I watch less television. (Chain Rule) 43 If I like school, I won’t get a job after school. (Chain Rule) 44 1. a 2. p ∨ t 3. a → ~t 4. ~t Law of Detachment (3, 1) 5. p Law of Disjunctive Inference (2, 4) 45 1. x → y 2. ~z ∨ x 3. z 4. x Law of Disjunctive Inference (2, 3) 5. y Law of Detachment (1, 4) 46 1. c ∨ a 2. ~c 3. a Law of Disjunctive Inference (1, 2) 4. a → b 5. b Law of Detachment (4, 3) 47 1. c → ~d 2. d 3. ~c Law of Modus Tollens (1, 2) 4. ~b → c 5. b Law of Modus Tollens (4, 3) 6. a ∨ ~b 7. a Law of Disjunctive Inference (6, 5) 48 1. ~m ∨ n 2. ~m → r 3. r → ~q 4. ~m → ~q Chain Rule (2, 3) 5. q 6. m Law of Modus Tollens (4, 5) 7. n Law of Disjunctive Inference (1, 6) 8. n → z 9. z Law of Detachment (8, 7) Chapter Review 7
CHAPTER 3 3-1 Inductive Reasoning (page 42) 1 For a polygon with n sides and vertices, from each vertex, diagonals can be drawn to n 3 other vertices. This can happen n times. Each diagonal is counted twice from a to b and n(n 3) from b to a. There are _ 2 diagonals. Test: Triangle 0 Quadrilateral 4(4 3) _ 2 2 5(5 3) Pentagon _ 5 2 2 n 5 1 2 3 4 5 5 4 3 2 1 −−−−−−−−−−− 6 6 6 6 6 5(6) n(n 1) twice the sum The sum of the integers from 1 through n(n 1) n _ 2 . 3 Answers will vary. 9, 15, 21 are some possible answers. 4 Answers will vary. 3, 9, 15, 21 are some possible answers. 3-2 Definitions and Logic (page 43) 1 Line segments are congruent if and only if they have the same measure. 2 A point of a line segment is the midpoint if and only if it divides the line segment into two congruent segments. 8 Chapter 3: Introduction to Geometric Proof Introduction to Geometric Proof 3 A line is an angle bisector if and only if it divides an angle into two congruent parts. 4 Lines are perpendicular if and only if the lines form right angles. 5 A figure is a polygon if and only if it is a closed figure in the plane formed by three or more segments joined at their endpoints. 3-3 Deductive Reasoning (page 44) 1 1. Given 2. Perpendicular lines form right angles. 3. Right triangles have one right angle. 2 EG divides FEH into two adjacent angles with the same measure. FEG and HEG are congruent because they have the same measure. An angle bisector is the ray that divides an angle into two adjacent congruent angles. 3 1. Given 2. Line segments equal in measure are congruent. 3. A midpoint divides a line segment into two congruent segments. 4 Since −− PR bisects SPQ, it divides the angle into two congruent angles. SPR QPR. If two angles are congruent, they have the same measure.
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: 3 1. e ∨ ~f 2. ~f → g 3. ~e 4.
Page 13 and 14: 6. BAC DAE 6. Transitive property.
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
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7. MAD RCB 8. MAD RCB (SAS SAS)
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5. RSQ TSV (Vertical angles) 6. QR
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3 4x 2 3x 3 x 5 RS 18 4 Perime
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Note: Since there are many variatio
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28 Enclose PAT in a large rectangle
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Geometry of Three Dimensions 11-1 P
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11-6 Volume of a Prism (pages 269-2
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14 47.5 in. 2 15 h 4 in. 16 25 cm
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14 a 3 : 2 b QR 10, RS 20, ST 12
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4. A A 5. ADE ABC b (AA) AC _ AB
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12 7 √ 2 13 4 14 5 √ 3 15 x
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13-2 Arcs and Chords (pages 350-351
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27 1. Common external tangents, −
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8. mHCT 1 _ m 2 TH mCHT 1 _ CT 2
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16 a (2 √ 2 , 2 √ 2 ), (2 √
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15 a Use constructing a congruent a
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5 Two horizontal lines y 11 and y
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Each review has a total of 58 possi
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Part III For each question, use the
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Chapters 1-3 (pages 438-441) Part I
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Part IV For each question, use the
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16 Score Explanation 4 a 91, b 35
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20 Score Explanation 6 The followin
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Part II For each question, use the
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Part III For each question, use the
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12 Score Explanation 2 31, and an a
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16 Score Explanation 4 DN √ 106
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20 Score Explanation 2 a 3, and an
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17 Score Explanation 4 6 √ 5 , a
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16 Score Explanation 4 mADE 93, mB
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20 Score Explanation Chapters 1-14
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Practice Regents Examination One (p
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36 Score Explanation 4 36, and an a
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32 Score Explanation 2 108, and an
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Part II For each question, use the
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