Preparing for the Regents Examination Geometry, AK

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22 Label arcs with degree measures that sum to 360, as with x, 2x, 3x, 4x. Then find the values of the inscribed angles. 23 Parallel lines cut congruent arcs x and y. Therefore, 2x 2y 360 x y 180 Each inscribed angle is equal to one-half the intercepted arc: 90. 13-4 Tangents and Secants (pages 363–365) 1 a b c d 2 a 12 b 3 3 a 3.25 b 1 4 a disjoint b externally tangent c intersecting twice d internally tangent e disjoint internally f concentric 5 a Sketch of two circles externally disjoint b Sketch of two intersecting circles, not tangent c Sketch of externally tangent circles d Sketch of internally tangent circles 6 a 1 b 3 c 2 d 4 7 a 160 b 130 c 90 d 40 e 180 x 8 a 60 b 45 c 35 d 51 e 67.5 180 n 9 a _ b 90 2n c n 2 _ 2 d 45 n e __ 180 m n 2 10 16 11 80 12 48 13 234 14 a 18 b 72 c 29.25 15 a 11, 17, 18 b no sides equal 16 a 12, 16, 20 b Satisfies Pythagorean theorem: 12 2 16 2 20 2 17 a 10, 15, 15 b two sides are equal 18 BE 4, EC 5, CF 5, AF 6, AC 11, AB 10 19 AB 20, CB 20, RB 10 20 AB 24, DR 14, DB 32 21 OB 26, DB 36, RB 16 22 OB 20, AB 2 √ 91 , CB 2 √ 91 23 OB 22, CB 8 √ 6 , AB 8 √ 6 24 AB 20, CB 20, OC 15, OB 25 25 Tangent segments from the same point are congruent. Base angles are equal, each measuring 60. Therefore, APB is equilateral. 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.) 26 1. Let E be the point of intersection of −− AB and −−− CD . 2. −− EB and −− ED are tangent segments to circle P. 3. −− EB −− ED (Two tangent segments drawn from an external point are congruent.) 4. −− EA and −− EC are tangent segments to circle O. 5. −− EA −− EC 6. −− EA −− EB ( Addition postulate) −− EC −−− CD or −− AB −−− CD 13-4 Tangents and Secants 81
27 1. Common external tangents, −−− AG and −−− CM 2. Radii −−− OA and −−− OC 3. OAG and OCM are right angles. 4. OAG and OCM are right triangles. 5. −−− OA −−− OC 6. Tangents −−− OM and −−− OG of circle P 7. −−− OM −−− OG (Two tangent segments drawn from an external point are congruent.) 8. OAG OCM (HL HL) 9. −−− AG −−− CM (CPCTC) 28 1. Circles A and B are congruent with tangent FG . 2. −− AF is the radius of A. −− BG is the radius of B. 3. −− AF −− BG 4. AFC and BGC are right angles. 5. FCA GCB 6. FCA GCB (AAS AAS) 7. −− AC −− BC 29 mA 1 _ ( 2 CD BE ). 2(mA) mCOD mBOE. But mBOE mA. 2(mA) mCOD mA. Therefore, 3(mA) mCOD. 30 1. −− AB is tangent to O at E; −−− CD is tangent to O at F. 2. −− OE OF 3. −−− OA −−− OC ; −− OB −−− OD 4. −− OE −− AB 5. OEA and OEB are right angles. 6. −− OF −−− CD 7. OFC and OFD are right angles. 8. OEA OFC (HL HL) OEB OFD 9. −− AE −− EB (Addition −− CF −− FD postulate) or −− AB −−− CD 13-5 Angles Formed by Tangents, Chords, and Secants (pages 368–371) 1 (2) 96 2 (1) 24 3 (2) 144 82 Chapter 13: Geometry of the Circle 4 (4) 173 5 (1) 38 6 (1) 90 7 a 40 b 62 c 225 d 45 e 75 f 100 g 80 h 30 8 a 79 b 38 c x 50, y 60, z 120 9 a 66 b 80 c 73 d 130 e 33 f 73 10 a 90 b 120 c 90 d 60 e 30 f 60 g 30 h 90 11 a 120 b 40 c 80 d 40 e 140 f 100 12 BC CD because in a regular hexagon, congruent chords subtend congruent arcs. Chords and a tangent that intercept congruent arcs are parallel. Therefore, PC −− BD 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.) 13 1. −− FB −− EC 2. BEC EBF 3. EHJ BHG 4. EHJ BHG (Alternate interior angles are congruent.) 5. BH _ HE HG _ HJ (Corresponding sides of similar triangles are in proportion.) 6. BH JH (The product of means HE HG equals the product of 14 1. extremes.) −− BA −−− DA 2. Diameter −−− DC 3. BAD and DBC are right angles. 4. −− AB is tangent to O at B. 5. mDBA 1 _ m 2 BD 6. mDCB 1 _ m 2 BD 7. mDBA mDCB 8. DBA DCB 9. DBA DCB
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ANSWER KEY Preparing for the REGENT
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Contents Chapter 1: Essentials of G
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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.
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CHAPTER 3 3-1 Inductive Reasoning (
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6. BAC DAE 6. Transitive property.
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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
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5-3 Isosceles and Equilateral Trian
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12. −−− HG −−− DC 13.
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5 1. −− FG is the perpendicular
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Chapter Review (pages 84-85) Note:
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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 and 76: 14 47.5 in. 2 15 h 4 in. 16 25 cm
Page 77 and 78: 14 a 3 : 2 b QR 10, RS 20, ST 12
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: 13-2 Arcs and Chords (pages 350-351
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
Page 127 and 128: 16 Score Explanation 4 DN √ 106
Page 129 and 130: 20 Score Explanation 2 a 3, and an
Page 131 and 132: 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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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
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Part II For each question, use the
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Part IV For each question, use the
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