Preparing for the Regents Examination Geometry, AK

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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.) 5 1. −−−− CMD is a line segment. 2. −−− CM −−− MD 3. M is the mid- (Definition of a midpoint) point of −−− CD . 6 1. −−−− ABCD is a line segment. 2. −− AB −−− CD 3. BC BC 4. AB CD 5. AB BC CD BC 6. AC BC 7. −− AC −− BD (Line segments that are equal in measure are congruent.) 7 1. −− AB −−− CD 2. AB CD 3. −− AB and −−− CD bisect each other at E. 4. AE 1 _ AB 2 5. CE 1 _ CD 2 6. AE CE 7. −− AE −− CE (Line segments that are equal in measure are congruent.) 8 1. −− PQ −−− QR 2. mPQR 90 3. mPQR mABC 4. mABC 90 5. −− AB −− BC (Perpendicular lines form right angles.) 9 1. PQ XY 2. QR YZ 3. PQ QR XY YZ 4. PQ QR PR 5. XY YZ XZ 6. PR XZ 7. −− PR −− XZ (Line segments that are equal in measure are congruent.) 10 1. DAB and EBA are straight angles. 2. mDAB 180 3. mEBA 180 4. m1 m2 5. mDAB m1 mEBA m2 6. m3 m4 (Substitution postulate) 11 1. −− AC −− BC 2. AC BC 4. −−− DC −− EC 5. DC EC 6. AD DC BC EC 7. AD BE 8. −−− AD −− BE (Line segments that are equal in measure are congruent.) 12 1. −−− AD −− BC 2. AD BC 3. E is the midpoint of −−− AD . 4. DE 1 _ AD 2 5. F is the midpoint of −− BC . 6. BF 1 _ BC 2 7. BF DE 8. −− BF −− DE (Line segments that are equal in measure are congruent.) Chapter Review 11
CHAPTER 4-1 Setting Up a Valid Proof (pages 56–57) 1 Given: ABC , BD AC D 4 A C B Prove: ABD CBD Statements Reasons 1. ABC 2. BD AC 1. Given. 2. Given. 3. ABD is a right angle. 4. CBD is a right angle. 3. Definition of perpendicular lines. 4. Definition of perpendicular lines. 5. ABD CBD 5. Congruent angles are angles that have the same measure. 2 Given: PQRS , −− PR −− QS P Q R S Prove: −− PQ −− RS 12 Chapter 4: Congruence of Lines, Angles, and Triangles Congruence of Lines, Angles, and Triangles Statements Reasons 1. PQRS 1. Given. 2. −− PR −− QS 2. Given. 3. −−− QR −−− QR 3. Reflexive property of congruence. 4. PR QR QS QR 4. Subtraction postulate 5. −− PQ −− RS 5. Substitution postulate 3 Given: −−− AD −− BC −− EF bisects −−− AD −− GF bisects −− BC Prove: −− AE −− BG Statements Reasons 1. −−− AD −− BC 1. Given. 2. −− EF bisects −−− AD 2. Given. 3. −− GF bisects −− BC 3. Given. 4. −− AE −−− AD 4. Definition of a bisector. 5. −− BG −− GC 5. Definition of a bisector. 6. AE ED 6. Definition of congruent segments. 7. BG GC 7. Definition of congruent segments. 8. AE ED AD 8. Partition postulate.
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 and 12: CHAPTER 3 3-1 Inductive Reasoning (
Page 13: 6. BAC DAE 6. Transitive property.
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)
Page 63 and 64: 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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