Preparing for the Regents Examination Geometry, AK

More documents

Recommendations

Info

9. BG GC BC 9. Partition postulate. 10. AE AE AD or 2AE AD 10. Substitution postulate. 11. BG BG BC or 11. Substitution 2BG BC postulate. 12. AE 1 _ AD 2 12. Division postulate. 13. BG 1 _ BC 2 13. Division postulate. 14. AD BC 14. Definition of congruent segments. 15. AE 1 _ BC 2 15. Substitution postulate. 16. 1 _ BC BG 2 16. Symmetric property. 17. AE BG 17. Transitive property. 18. −− AE −− BG 18. Segments with equal measures are congruent. 4 Given: URS UTS −− US bisects RUT −− US bisects RST Prove: 1 2 Statements Reasons 1. URS UTS 1. Given. 2. −− US bisects RUT 2. Given. 3. −− US bisects RST 3. Given. 4. 1 TUS 4. Definition of an angle bisector. 5. RSU 2 5. Definition of an angle bisector. 6. m1 mTUS 6. Congruent angles are equal in measure. 7. mRSU m2 7. Congruent angles are equal in measure. 8. m1 mTUS 8. Partition mRUT postulate. 9. mRSU m2 9. Partition mRST postulate. 10. m1 m1 mRUT or 2(m1) mRUT 11. m2 m2 mRST or 2(m2) mRST 12. m1 1 _ 10. Substitution postulate. 11. Substitution postulate. mRUT 12. Division 2 postulate. 13. m2 1 _ mRST 13. Division 2 postulate. 14. mRUT mRST 14. Congruent angles are equal in measure. 15. m1 1 _ 2 mRST 15. Substitution postulate. 16. 1 _ mRST 2 16. Symmetric 2 property. 17. m1 m2 17. Transitive property. 18. 1 2 18. Congruent angles are equal in 5 Given: measure. −− HJ −− FK Prove: −−− HK −− FJ Statements Reasons 1. −− HJ −− FK 1. Given. 2. HJ FK 2. Definition of congruent segments. 3. JK JK 3. Reflexive property. 4. HJ JK HK 4. Partition postulate. 5. FK JK FJ 5. Partition postulate. 6. HJ JK FK JK 6. Addition postulate. 7. HK FJ 7. Substitution postulate. 8. −−− HK −− FJ 8. Segments with equal measures are congruent. 4-1 Setting Up a Valid Proof 13
6 Given: BA bisects CBE 1 2 2 4 Prove: 3 4 Narrative Proof: A bisector divides an angle into two congruent angles, so 1 2. Using the transitive property of congruence, 1 4. By substitution, 3 4. 7 4x 3 2x 21 x 12 mBCA 4x 3 4(12) 3 45 mBCE 2(45) 90 8 4x 4 2(3x 21) x 23 AB 4x 4 4(23) 4 96 9 Since DE CE, then 2x y 5y, and x 2y. AC BD 6 5y (2x y) (x y) 6 5y 3x 2y 6 5y 3(2y) 2y 6 5 y 6y 2y 6 3y 2 y x 2(2) 4 BD 2(4) 2 4 2 16 10 AB DE 8, so BD 4 and CD 2. Therefore, CE 8 2 10. 4-2 Proving Theorems About Angles (pages 59–61) 1 If two angles are right angles, they both have a measurement of 90. Angles with the same measure are congruent. 2 If two angles are straight angles, they both have a measurement of 180. Angles with the same measure are congruent. 3 If 1 is a complement of A, then m1 90 mA. If 2 is a complement of A, then m2 90 mA. Then m1 m2 by the symmetric property and transition, and 1 2 by definition of congruence. 14 Chapter 4: Congruence of Lines, Angles, and Triangles 4 If 1 2, 3 is the complement of 1, and 4 is the complement of 2, then m3 90 m1 and m4 90 m2. Since m1 m2 by congruence, m3 m4 by symmetric property and transition, and 3 4 by congruence. 5 If 1 is a supplement of A, then m1 180 mA. If 2 is a supplement of A, then m2 180 mA. Then m1 m2 by the symmetric property and transition, and 1 2 by definition of congruence. 6 If 1 2 and 3 is the supplement of 1 and 4 is the supplement of 2, then m3 90 m1 and m4 90 m2. Since m1 m2 by congruence, m3 m4 by symmetric property and transition, and 3 4 by congruence. 7 By definition the sum of their measures is a straight angle or 180. 8 The two angles are a linear pair so the sum of their measures is 180. If they are congruent, each measure is 90; they form right angles and are therefore perpendicular. 9 Each angle forms a linear pair with the same angle. They are supplementary to the same angle and are therefore congruent. 10 m2 m3 because they are vertical angles. By substitution, 1 is complementary to 3. 1 4 because complements of the same angle are congruent. 11 3 forms a linear pair with 1, 3 is supplementary to 1. Since 1 2, 3 is supplementary to 2. 4 forms a linear pair with 2, 4 is supplementary to 2. 3 4 because supplements of the same angle are congruent. 12 Since they form a linear pair, 2 is supplementary to EDG. Since 1 and 2 have the same measure (they are congruent), 1 is supplementary to EGD. 13 Since 1 is complementary to 3, m1 m3 90. Since ACB is a right angle, m1 m2 90. By subtraction and substitution, m2 m3, so they are congruent angles. 14 By the transitive postulate, 1 3. Because they are vertical angle pairs, 1 4, and 3 6. By substitution and transition, 4 6.
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 and 14: 6. BAC DAE 6. Transitive property.
Page 15: CHAPTER 4-1 Setting Up a Valid Proo
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 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 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:
Page 107 and 108:
Page 109 and 110:
20 Score Explanation 6 The followin
Page 111 and 112:
Page 113 and 114:
Part II For each question, use the
Page 115 and 116:
Page 117 and 118:
Part III For each question, use the
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
12 Score Explanation 2 31, and an a
Page 125 and 126:
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 133 and 134:
Page 135 and 136:
16 Score Explanation 4 mADE 93, mB
Page 137 and 138:
20 Score Explanation Chapters 1-14
Page 139 and 140:
Page 141 and 142:
Practice Regents Examination One (p
Page 143 and 144:
36 Score Explanation 4 36, and an a
Page 145 and 146:
32 Score Explanation 2 108, and an
Page 147 and 148:
Part II For each question, use the
Page 149:
show all

Preparing for the Regents Examination Geometry, AK

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?