Preparing for the Regents Examination Geometry, AK

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10-6 Squares (pages 232–233) 1 (4) A rectangle is a square. 2 (3) sides and angles are congruent 3 (4) A trapezoid is a parallelogram. 4 (3) −− AC −−− DC 5 (1) congruent and bisect the angles to which they are drawn 6 (2) x √ 2 7 Slope of −−− DA slope of −− VE 3 _ . Slope of 4 −− ED slope of −− AV 4 _ . A parallelogram 3 has two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides have slopes that are negative reciprocals. Slope of −−− DV 1 _ . Slope 7 of −− AE 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular. Therefore, DAVE is a square. 8 Slope of −−− MA slope of −− TH 4 _ . Slope of 3 −−− HM slope of −− AT 3 _ . A parallelogram has 4 two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides have slopes that are negative reciprocals. Slope of −−− MT 1 _ . Slope of 7 −−− AH 7. Slopes of the diagonals are negative reciprocals, thus they are perpendicular. Therefore, MATH is a square. 9 The diagonals bisect the vertex angles, creating four congruent isosceles triangles. The bisected angles (the base angles of the triangles) measure 45. The vertex angles thus measure 90. The diagonals cross, forming 90 angles. 10 a Slope of −− PQ slope of −− RS 0. −−− QR and −− SP have no slope. A parallelogram has two pairs of opposite sides that are parallel. A rectangle is a parallelogram in which consecutive sides are perpendicular. b PQ RS 20. QR PS 7. Since all sides are not congruent to each other, PQRS is not a square. 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.) 11 a Given ABCD is a square. BFE is a right angle because −− EF −− BD . mEBF 45 because diagonal −− BD bisects right angle ABC. mFEB 45. Therefore, −− BF −− EF . b 1. ABCD is a square. 2. −− EF −− BD 3. EFD is a right angle. 4. DAE is a right ( Definition of angle. a square) 5. DAE and DFE are right triangles. 6. (Draw −− DE ). −− DE −− DE 7. DAE DFE (HL HL) 8. −− EF −− EA (CPCTC) 12 1. Rhombus ABCD with diagonals −− AC and −− BD intersecting at E. 2. 1 2 3. −− BE −− CE 4. E bisects −− AC and −− BD . 5. −− BD −− AC 6. ABCD is a rectangle. 7. ABCD has all right angles. 8. ABCD is a square. (A square is a rhombus in which all angles are right angles.) 10-7 Trapezoids (pages 236–238) 1 (2) They are congruent. 2 (3) ADC ABC 3 a x z 70; y 110 b x 73; y z 107 c x 40; y 108; z 32 d x 70; y 44; z 66 e x 130; y 20; z 30 f x 82; y z 41 4 Slope of −−− MA slope of −− TH 1 _ . These legs 2 are parallel. Slope of −− AT 3 _ HM has 4 . −−− no slope. These legs are not parallel. AT HM 10. Therefore, MATH is an isosceles trapezoid. 10-7 Trapezoids 63
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. ABCD is an isosceles trapezoid. 2. BAD CDA 3. −− BC −−− AD 4. NBC BAD 5. NCB CDA (Corresponding angles are congruent.) 6. NBC NCB (Substitution postulate) 7. NBC is isosceles. (Base angles of an isosceles triangle are congruent.) 6 1. Isosceles trapezoid ABCD 2. −− AB −−− DC 3. BAD CDA (Base angles of an isosceles trapezoid 4. are congruent.) −−− AD −−− AD 5. ADB DAC 7 1. Trapezoid ABCE 2. (SAS SAS) −− BD −−− AD 3. −− AB −− CE 4. A B 5. CED A 6. ECD B (Corresponding angles are congruent.) 7. CED ECD (Substitution 8. postulate) −−− CD −− ED 9. −− BD −−− CD ( Subtraction −−− AD −− ED 10. postulate) −− BC −− AE 11. ABCE is an (Nonparallel sides isosceles trapezoid. of an isosceles trapezoid are congruent.) 8 1. Quadrilateral PQRS 2. −−− QAB , −−− RAS , and −−− PSB 3. −− QB bisects −− RS . 4. −− RA −− AS 64 Chapter 10: Quadrilaterals 5. −−− PSB −−− QR 6. RAQ BAS (Vertical angles are congruent.) 7. QRA BSA (Alternate interior angles are congruent.) 8. QRA BSA 9. (ASA ASA) −−− QA −− AB 9 1. Isosceles trapezoid ABCD 2. A D 3. −−− AD −− BC 4. −− AB −−− DC 5. E is the midpoint of −−− AD . 6. −− AE −− ED 7. ABE DCE 8. a (SAS SAS) −− BE −− CE 9. BE CE (CPCTC) 10. BCE is isosceles. (Definition of isos- 11. b celes triangle) −− EH −− BC (The median from the vertex angle of an isosceles triangle is perpendicular to the base.) 10 1. Isosceles trapezoid PQRS 2. −− PQ −− RS 3. −−− QR −− PS 4. P S 5. −− PS −− PS 6. PQS SRP (SAS SAS) 7. PQS SRP (CPCTC) 8. QAP BAS (Vertical angles are congruent.) 9. PAQ SAR 11 1. Trapezoid ABCD 2. (AAS AAS) −− BR −−− AD 3. −−− CM −−− AD 4. ARB and DMC are right angles. 5. ARB and DMC are right triangles. 6. −− AB −−− CD 7. BAR CDM 8. ARB DMC (Hypotenuse–acute angle) 9. 1 2 (CPCTC) 12 1. Trapezoid PQRS 2. Q R 3. −− PA −−− QR , −− SE −−− QR 4. PAQ and SER are right angles. 5. PAQ and SER are right triangles.
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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.
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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
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
Page 65: 3 4x 2 3x 3 x 5 RS 18 4 Perime
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 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 105 and 106: Chapters 1-5 (pages 446-449) Part I
Page 109 and 110: 20 Score Explanation 6 The followin
Page 111 and 112: 17 Score Explanation 4 The followin
Page 113 and 114: Part II For each question, use the
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Part III For each question, use the
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Chapters 1-9 (pages 461-463) Part I
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19 Score Explanation 6 The followin
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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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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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