Preparing for the Regents Examination Geometry, AK

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43 1. −−− MA −−− OK , −−− MD −− OL 2. OAM and ODM are right angles. 3. −−− AM −−− DM 4. −−− OM −−− OM 5. OAM ODM (HL HL) 6. AOM LOM (CPCTC) 7. MK ML (Congruent angles have congruent arcs.) Locus and Constructions 14-1 Basic Constructions (page 404) Note: For exercises 1–20, check students’ constructions; procedures may vary. 1 ab Use construction of congruent angles procedure. 2 Use construction of a perpendicular to a line through a given point on the line procedure. 3 ac Use construction of an angle bisector procedure. 4 Use construction of a line perpendicular to a line through a given point not on the line procedure. 5 Use construction of a perpendicular bisector of a line segment procedure. 6 Use construction of a parallel line through a given point not on the line procedure. 7 Use construction of the median of a triangle procedure. 8 Use construction of line perpendicular to a line through a given point not on the line procedure. 44 1. −− OE −− AB , OD −− AC 2. AEO and ADO are right angles. 3. −− AB −− AC 4. −− OE bisects −− AB . 5. −−− OD bisects −− AC . 6. −− AE −−− AD 7. −−− OA −−− OA 8. AEO ADO (HL HL) 45 Use equal arcs are subtended by equal chords. CHAPTER 14 9 Use construction of a perpendicular bisector procedure. 10 Use construction of an angle bisector procedure. 11 a Use construction of a perpendicular line procedure. b Use construction of a perpendicular line procedure and then construct an angle bisector. 12 a Use construction of an equilateral triangle procedure. b Using angle constructed in a, construct an angle bisector. c Construct angle bisector of b. 13 Using the vertex of the angle as the center of a circle and one of the legs as its radius, construct a circle. Construct the diameter by extending the radius. 14 Using the vertex of the angle as the center of a circle and one of the legs as its radius, construct a circle. Construct the diameter by extending the radius. Construct a perpendicular through the center of the circle. 14-1 Basic Constructions 87
15 a Use constructing a congruent angle procedure. b Use constructing a congruent angle procedure. c Construct a perpendicular bisector of a segment to form a 90 angle. Then use constructing a congruent angle procedure to construct the sum. d Use constructing an angle bisector procedure. e Use constructing an angle bisector procedure for A. Then use constructing a congruent angle procedure to construct the sum. 16 Use construction of an equilateral triangle procedure. 17 Use the construction of an equilateral triangle procedure using the length of the longer segment to construct the leg of the isosceles triangle. 18 Using the compass, measure the radius and construct a circle passing through the point on the tangent line. 19 Use construction of a line tangent to a given circle through a given point outside the circle procedure. 20 Use procedures for constructing parallel lines and perpendicular lines. 14-2 Concurrent Lines and Points of Concurrency (pages 407–408) 1 (4) obtuse 2 (3) at one of the vertices of the triangle 3 Check students’ constructions. 4 All at the same point 5 10 6 6 2 _ 3 7 36 8 16.5 9 4.5 10 SP 4, SC 6 11 PB 15, PR 30 12 PA 15, QA 45 13 4 1 _ 3 14 9 88 Chapter 14: Locus and Constructions 15 3 √ 3 16 QE 4, DE 6 17 a, b, c: all interior 18 a, b, c: all interior 19 a interior b on a side c exterior 20 a interior b at a vertex c exterior 21 Incenter: √ 3 , circumcenter: 2 √ 3 22 Incenter: 2 √ 3 , circumcenter: 4 √ 3 23 Incenter: 3 √ 3 , circumcenter: 6 √ 3 24 Incenter: 6, circumcenter: 12 25 Since AG GC, the base is the same and the altitude is the same. Therefore, the area is the same. 14-4 Six Fundamental Loci and the Coordinate Plane (pages 413–414) 1 (3) One concentric circle of radius 11 inches 2 (2) one line 3 (2) a point 4 (1) a circle of radius 4 with center at P 5 (4) a circle Note: For exercises 6–17 check students’ sketches. 6 Circle with given point as center and radius 3 7 Two parallel lines, one on each side of the given line 8 One line parallel to and midway between the given parallel lines 9 The perpendicular bisector of the segment joining R and S 10 The perpendicular bisector of the segment AB 11 A circle with radius 2.5 inches and the given point as the center 12 The line that is the bisector of ABC 13 Two lines each the bisector of the vertical angles 14 Two concentric circles with radii 5 and 9 15 One concentric circle midway between the given circles 16 All the points in the interior of a circle with the given point as the center and with a radius 2 inches 17 A circle with the given point as the center and a radius of 3 inches and all the points in the exterior of that circle
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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
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6-3 Line Reflections and Symmetry (
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11 A(0, 8), B(2, 2), C(6, 4) (8) 10
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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 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: 16 a (2 √ 2 , 2 √ 2 ), (2 √
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
Page 135 and 136: 16 Score Explanation 4 mADE 93, mB
Page 137 and 138: 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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Part IV For each question, use the
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Preparing for the Regents Examination Geometry, AK

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