Preparing for the Regents Examination Geometry, AK
Preparing for the Regents Examination Geometry, AK Preparing for the Regents Examination Geometry, AK
12. BAC EDF (CPCTC) 13. ___ ED ___ AD (Alternate interior angles are congruent) 14 1. ___ BE bisects ABC. 2. 1 EBC 3. m1 mEBC 4. ___ CE bisects DCB. 5. 2 ECB 6. m2 mECB 7. m1 m2 90 8. mEBC mECB 90 (Substitution postulate) 9. m1 m2 mEBC mECB 10. m1 mEBC 90 11. m2 mECB 90 12. ABC and DCB are right angles. 13. BA ___ BC 14. CD ___ BC 15. BA CD (Two lines perpendicular to the same line are parallel.) 15 1. 1 3 2. m1 m3 3. 2 4 4. m2 m4 5. m1 m2 m3 m4 360 6. m1 m1 m2 m2 360 or 2(m1) 2(m2) 360 7. m1 m2 180 (Division postulate) 8. −−− AD −− BC (Interior angles on the same side of the transversal are supplementary.) 9. m1 m1 + m4 m4 360 or 2(m1) 2(m4) 360 10. m1 m4 180 11. ___ AB ___ CD 16 1. D is the midpoint of ___ CF and of ___ BE . 2. ___ CD ___ DF 3. ___ BD ___ DE 4. ___ FC ___ FC 5. CBD FDE (Vertical angles are congruent.) 6. CBD FDE (SAS SAS) 7. EFD DCB (CPCTC) 8. ___ AC ___ FE (Alternate interior angles are congruent.) 17 1. 1 2 2. AFC DCF (Supplementary angles of congruent angles are congruent.) 3. ___ EF ___ CB 4. ___ FC ___ FC 5. ___ EF ___ FC = ___ BC ___ FC 6. ___ EC ___ BF 7. ___ AF ___ CD 8. AFB DCE (SAS SAS) 9. ABF DEC (CPCTC) 10. ___ AB ___ ED (Alternate interior angles are congruent.) 18 1. ___ AE ___ FC 2. ___ EF ___ EF 3. ___ AE ___ EF ___ FC ___ EF 4. ___ AF ___ CE 5. ___ DE ___ AC 6. DEA is a right angle. 7. ___ BF ___ AC 8. BFC is a right angle. 9. DEA BFC (Right angles are congruent.) 10. ___ DE ___ BF 11. AFB CED (SAS SAS) 12. ______ AEFC 13. BAF DCE (CPCTC) AB ___ DC 14. ___ 9-3 Properties of Parallel Lines (pages 182–185) 1 (1) same-side exterior angles 2 (2) 2, 3, 6, 7, 10, 11, 14, 15 3 m1 m4 m6 m7 60; m2 m3 m5 120 4 m1 m4 m5 m8 135; m2 m3 m6 m7 45 5 ma md mg 65; mb mc me mf 125 6 w y 70; x z 110 7 a 8x 6x 3 0 180 x 15 b 2x 1 0 5x 47 x 19 m3 2(19) 10 48 9-3 Properties of Parallel Lines 45
8 106 9 mA 75, mC 67 10 57 11 60 12 1. Perpendicular lines form right angles. 2. Right angles measure 90. 3. When two parallel lines are cut by a transversal, corresponding angles are congruent. 4. Congruent angles have equal measure. 5. Transitive property of congruence (3, 5) 6. If an angle measures 90, it is a right angle. 7. If two lines intersect to form a right angle, they are perpendicular. 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. ___ AB 2. 2 is supplement of 1. 3. 2 is supplement of 3. 4. 1 3 (If two angles are supplements of the same angle, then they are congruent.) 5. ___ BC ___ AD (When two lines are cut by a transversal creating corresponding angles, the lines are congruent.) 14 1. ABC 2. ___ AB ___ BC 3. BAC BCA (Isosceles triangle theorem) 4. _____ FHD ___ BC 5. FDA BCA (Corresponding exterior angles are congruent.) 6. _____ EHG ___ AB 7. BAC GEC (Corresponding exterior angles are congruent.) 8. GEC FDA 9. ___ HE ____ HD 10. EHD is isosceles. (Definition of isosceles triangle) 46 Chapter 9: Parallel Lines 15 1. ___ DE ___ AC 2. 1 2 (Corresponding exterior angles are congruent.) 3. 3 4 4. 2 3 5. 1 4 (Transitive property of congruence) 16 1. ___ AC intersects ___ BD at E. 2. ___ AE ___ ED 3. A D 4. ___ AD ___ BC 5. A C (Alternate interior angles are congruent.) 6. B D (Alternate interior angles are congruent.) 7. B C (Transitive property of congruence) 8. ___ AE ___ CE (Isosceles triangle theorem) 9. ___ BE ___ DE 10. ___ AC ___ BD (Addition postulate) 17 1. n m 2. ABE and BAD are supplementary. (Two interior angles on the same side of the transversal are supplementary.) 3. mABE mBAD 180 4. 1 _ mABE 2 1 _ mBAD 90 2 (Division postulate) 5. ___ BC bisects ABE. 6. mABC 1 _ mABE 2 (Definition of angle bisector) 7. ___ AC bisects BAD. 8. mACB 1 _ mBAD 2 ( Definition of angle bisector) 9. mABC mACB 90 10. mBCA mABC mACB 180 11. mBCA 90 12. BCA is a right angle. 13. ___ BC ___ AC (Perpendicular lines form right angles.) 18 1. r m and a b 2. 4 and 3 are supplementary. 3. 2 and 3 are supplementary. 4. (a) 4 2 (Supplements of the same angle are congruent.) 5. 4 and 5 are supplementary.
- 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 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: ___ 27 a M KA (5, 1), M ___ AT (4
- 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
12. BAC EDF (CPCTC)<br />
13. ___<br />
ED ___<br />
AD (Alternate interior angles are<br />
congruent)<br />
14 1. ___<br />
BE bisects ABC.<br />
2. 1 EBC<br />
3. m1 mEBC<br />
4. ___<br />
CE bisects DCB.<br />
5. 2 ECB<br />
6. m2 mECB<br />
7. m1 m2 90<br />
8. mEBC mECB 90 (Substitution<br />
postulate)<br />
9. m1 m2 mEBC mECB<br />
10. m1 mEBC 90<br />
11. m2 mECB 90<br />
12. ABC and DCB are right angles.<br />
13. BA ___<br />
BC<br />
14. CD ___<br />
BC<br />
15. BA CD (Two lines perpendicular to <strong>the</strong><br />
same line are parallel.)<br />
15 1. 1 3<br />
2. m1 m3<br />
3. 2 4<br />
4. m2 m4<br />
5. m1 m2 m3 m4 360<br />
6. m1 m1 m2 m2 360<br />
or 2(m1) 2(m2) 360<br />
7. m1 m2 180 (Division<br />
postulate)<br />
8. −−−<br />
AD −−<br />
BC (Interior angles on <strong>the</strong> same<br />
side of <strong>the</strong> transversal are<br />
supplementary.)<br />
9. m1 m1 + m4 m4 360<br />
or 2(m1) 2(m4) 360<br />
10. m1 m4 180<br />
11. ___<br />
AB ___<br />
CD<br />
16 1. D is <strong>the</strong> midpoint of ___<br />
CF and of ___<br />
BE .<br />
2. ___<br />
CD ___<br />
DF<br />
3. ___<br />
BD ___<br />
DE<br />
4. ___<br />
FC ___<br />
FC<br />
5. CBD FDE (Vertical<br />
angles are<br />
congruent.)<br />
6. CBD FDE (SAS SAS)<br />
7. EFD DCB (CPCTC)<br />
8. ___<br />
AC ___<br />
FE (Alternate interior angles are<br />
congruent.)<br />
17 1. 1 2<br />
2. AFC DCF (Supplementary angles<br />
of congruent angles are<br />
congruent.)<br />
3. ___<br />
EF ___<br />
CB<br />
4. ___<br />
FC ___<br />
FC<br />
5. ___<br />
EF ___<br />
FC = ___<br />
BC ___<br />
FC<br />
6. ___<br />
EC ___<br />
BF<br />
7. ___<br />
AF ___<br />
CD<br />
8. AFB DCE (SAS SAS)<br />
9. ABF DEC (CPCTC)<br />
10. ___<br />
AB ___<br />
ED (Alternate interior angles<br />
are congruent.)<br />
18 1. ___<br />
AE ___<br />
FC<br />
2. ___<br />
EF ___<br />
EF<br />
3. ___<br />
AE ___<br />
EF ___<br />
FC ___<br />
EF<br />
4. ___<br />
AF ___<br />
CE<br />
5. ___<br />
DE ___<br />
AC<br />
6. DEA is a right angle.<br />
7. ___<br />
BF ___<br />
AC<br />
8. BFC is a right angle.<br />
9. DEA BFC (Right angles are<br />
congruent.)<br />
10. ___<br />
DE ___<br />
BF<br />
11. AFB CED (SAS SAS)<br />
12. ______<br />
AEFC<br />
13. BAF DCE (CPCTC)<br />
AB ___<br />
DC<br />
14. ___<br />
9-3 Properties of Parallel<br />
Lines<br />
(pages 182–185)<br />
1 (1) same-side exterior angles<br />
2 (2) 2, 3, 6, 7, 10, 11, 14, 15<br />
3 m1 m4 m6 m7 60;<br />
m2 m3 m5 120<br />
4 m1 m4 m5 m8 135;<br />
m2 m3 m6 m7 45<br />
5 ma md mg 65;<br />
mb mc me mf 125<br />
6 w y 70; x z 110<br />
7 a 8x 6x 3 0 180<br />
x 15<br />
b 2x 1 0 5x 47<br />
x 19<br />
m3 2(19) 10 48<br />
9-3 Properties of Parallel Lines 45
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