Preparing for the Regents Examination Geometry, AK

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27 40, 50 28 70, 110 29 20, 35 30 16, 20, 24 31 length 28, width 49 32 45, 135 33 a 8 b 6 c 10 34 a 20 b 9 12-2 Proportions Involving Line Segments (pages 298–300) 1 a–d are all true 2 2 _ 5 4 _ Proportions are not true, lines not 8 parallel. 3 6 4 28 5 5.4 6 6 7 24 8 5 9 ac _ b 10 np _ m 11 6 12 20 c(a b) 13 _ a 14 No 15 No 16 No 17 No 18 Yes 19 12 20 1 : 3 or 1 _ 3 21 11 cm 22 x 6, AB 47, DE 17, AC 34, EC 9, BE 9, AD 23.5, DB 23.5 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.) 23 a 1. ABC 2. P is the midpoint of −− AB . 3. Q is the midpoint of −− BC . 4. −− PQ −− AC (A line joining the midpoints of two sides of a triangle is parallel to the third side.) 5. −− PQ ___ AR 6. R is the midpoint of −− AC . 7. −− AB −−− RQ 8. −− AP −−− RQ 9. PQRA is a parallelogram. (Definition of a parallelogram) b 30 24 1. ABC, ADC 2. −− wx −− AC (A line dividing two sides of a triangle proportionally is parallel to the third side.) 3. −− zy −− AC 4. −− wx −− zy 5. BAD, BCD 6. −− wz −− BD 7. −− xy −− BD 8. −− wz −− xy 9. wxyz is a ( Definition of a parallelogram. parallelogram) 25 48 √ 3 12-3 Similar Polygons (pages 302–303) 1 1 : 1 2 4, 4.5, 5 3 1 : 4 4 45, 55 5 a 2 : 3 b 4.5, 7.5 c 18 6 9, 13.5, 18 7 5a, 5b, 5c 8 All corresponding angles are congruent, all corresponding sides are in proportion, 1 : 2. 9 w _ , 3 x _ , 3 y _ , 3 z _ 3 10 32, 40 11 mR mQ 110 12 mI 40; mK 160 13 12 12-3 Similar Polygons 73
14 a 3 : 2 b QR 10, RS 20, ST 12, PT 22 15 a True b False c True d True e False f True g False h False 16 If the vertex angles are congruent, then the base angles must also be congruent. Similarly, if the base angles are congruent, then the vertex angles are congruent. Triangles are similar by (AA) or (AAA). 12-4 Proving Triangles Similar (pages 307–308) 1 (2) similar 2 Vertical angles are congruent. (AA) 3 Not similar 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.) 4 a 1. −− AE −− BC 2. BCD DAE ( Alternate interior angles) 3. −− AC and −− BE intersect at D. 4. BDC ADE (Vertical angles) 5. ADE CDB (AA) b AD 30 5 All corresponding sides are in proportion; AB _ BE CD _ AE CE _ 4 DE _ . Triangles 5 are similar. 6 4 _ and 6 5 _ , corresponding sides are not in 3 proportion. Triangles are not similar. 7 6 _ and 7 4 _ , corresponding sides are not in 5 proportion. Triangles are not similar. 8 1. −−− CD −− AB 2. CDA and CDB are right angles. 3. CDA CDB 4. CAD CBD 5. CAD CBD (AA) 74 Chapter 12: Ratios, Proportion, and Similiarity 9 1. −− AB −− DE 2. EDF BAC (Corresponding angles are congruent.) 3. −− BC −− EF 4. EFD BCA 5. ABC DEF (AA) 10 1. −− BE −− AC and −−− CD −− AB 2. HEC and HDB are right angles. 3. HEC HDB 4. DHB CHE (Vertical angles are congruent.) 5. EHC DHB (AA) 11 1. −− DE −− BC and −− DF −− AB 2. DEC and DFA are right angles. 3. DEC DFA 4. Parallelogram ABCD 5. FAD ECD 6. AFD CED (AA) 12 1. DBE ADF 2. BAC BDE (Corresponding angles) 3. B B 4. DBE ABC (AA) 13 1. Parallelogram ABCE 2. −−− AED 3. −− BC −− AE 4. −− BC −−− AD 5. CBF ADB (Alternate interior angles) 6. BAD FCB 7. BAD FCB (AA) 14 1. ABC PQR 2. ABC PQR 3. −− BD bisects ABC. 4. −− QS bisects PQR. 5. ABD PQS 6. BAD QPS 7. ABD PQS (AA) 15 1. Parallelogram ABCD 2. −−− AD −− BC 3. GAE BCG (Alternate interior angles are congruent.) 4. −− AC bisects HAE. 5. HAG GAE 6. HAG BCG 7. −−− AH −− AE
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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
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6 True 7 True 8 True 9 True 10 Answ
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3 1. e ∨ ~f 2. ~f → g 3. ~e 4.
Page 11 and 12:
CHAPTER 3 3-1 Inductive Reasoning (
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6. BAC DAE 6. Transitive property.
Page 15 and 16:
CHAPTER 4-1 Setting Up a Valid Proo
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6 Given: BA bisects CBE 1 2 2 4
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1 1. −−−− ABCD 2. 1 2 3. _
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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
Page 65 and 66: 3 4x 2 3x 3 x 5 RS 18 4 Perime
Page 67 and 68: Note: Since there are many variatio
Page 69 and 70: 28 Enclose PAT in a large rectangle
Page 71 and 72: Geometry of Three Dimensions 11-1 P
Page 73 and 74: 11-6 Volume of a Prism (pages 269-2
Page 75: 14 47.5 in. 2 15 h 4 in. 16 25 cm
Page 79 and 80: 4. A A 5. ADE ABC b (AA) AC _ AB
Page 81 and 82: 12 7 √ 2 13 4 14 5 √ 3 15 x
Page 83 and 84: 13-2 Arcs and Chords (pages 350-351
Page 85 and 86: 27 1. Common external tangents, −
Page 87 and 88: 8. mHCT 1 _ m 2 TH mCHT 1 _ CT 2
Page 89 and 90: 16 a (2 √ 2 , 2 √ 2 ), (2 √
Page 91 and 92: 15 a Use constructing a congruent a
Page 93 and 94: 5 Two horizontal lines y 11 and y
Page 95 and 96: Each review has a total of 58 possi
Page 97 and 98: Part III For each question, use the
Page 99 and 100: Chapters 1-3 (pages 438-441) Part I
Page 101 and 102: Part IV For each question, use the
Page 103 and 104: 16 Score Explanation 4 a 91, b 35
Page 109 and 110: 20 Score Explanation 6 The followin
Page 113 and 114: Part II For each question, use the
Page 117 and 118: Part III For each question, use the
Page 123 and 124: 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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20 Score Explanation 6 The followin
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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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Part IV For each question, use the
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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
Page 147 and 148:
Part II For each question, use the
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Part IV For each question, use the
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