Answers 49. ∠ FED
50. ∠ FDE
52. ∠ EDG
53. ∠ DFG
51. ∠ DFE
3. 3 7. 12
54. 11.4; Perpendicular Bisector Theorem (Thm. 6.1)
Theorem (Thm. 6.2) 57. 22; Perpendicular Bisector Theorem (Thm. 6.1) b. 36 units
8.1 Practice B
Chapter 8 8.1 Start Thinking
Sample answer: The three diagrams are the same image, but stretched or shrunk into different sizes or forms; The first resizing is not “similar” to the original in a geometric sense. The proportions of the map were not maintained. The second resizing is “similar” to the original in a geometric sense. It appears to be a dilation of the original in a geometric sense. It appears to be a dilation of the original by a factor less than one and maintains proportionally with the original.
3 ; ∠W ≅ ∠ S , ∠ X ≅ ∠ T , ∠ Y ≅ ∠ U , 2 WX XY YZ ZW = = = ∠ Z ≅ ∠V , ST TU UV VS
3. 6 7. a.
4. 9
5. 7 in.
6. 11 ft
7 4
c. 108°
3 2
2. x = 20
d. about 74.2 units e. about 219.73 square units
9 4. x = − 5
3. x = ± 3
5. x =
2.
3 ; ∠ A ≅ ∠ H , ∠ B ≅ ∠ I , ∠C ≅ ∠ J , 4 AB BC CA = = HI IJ JH
b. 7.5
8.1 Warm Up 1. x =
9. 9
12. 336 ft 2
11. 3
1.
c. 72 units
8. 60; 540
5.5), ADC ≅ BDC and XWZ ≅ YWZ . Because corresponding parts of congruent triangles are congruent, BC = 13 and YZ = 39.
Theorem (Thm. 6.2) 56. 36; Converse of the Perpendicular Bisector
6. 67°
5. 3
10. 13, 39; By the SAS Congruence Theorem (Thm.
55. 1.9; Converse of the Perpendicular Bisector
58. a. 3
4. 22.5
64 7
3 2
6. x = − , x = 4
8.1 Cumulative Review Warm Up 1. 120°
2. 60°
3. 60°
4. 60°
5. 75°
6. 45°
f. yes; Because corresponding angles of similar
triangles are congruent, ∠ ABC ≅ ∠ D. By the corresponding Angles Converse Theorem (Thm. 3.5), BC || DE. 8.1 Enrichment and Extension 1. Sample answer:
8.1 Practice A 1. 3; ∠ L ≅ ∠ Q , ∠ M ≅ ∠ R , ∠ N ≅ ∠ S ,
LM MN NL = = QR RS SQ 2.
2. Sample answer:
2 ; ∠ A ≅ ∠ E, ∠ B ≅ ∠ F , ∠ C ≅ ∠ G, 5 AB BC CD DA = = = ∠D ≅ ∠H, EF FG GH HE
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Geometry Answers
A85
Answers c. 50°
3. Sample answer:
d. 12 e. 12; Because the diagonals are perpendicular, by
the Rhombus Diagonals Theorem (Thm. 7.11), ABED is a rhombus. By definition, all sides are congruent. So, AD = 12. 8.1 Puzzle Time
6. no; m ∠ X = 45° and m ∠ C = 76°
HEY I THINK I AM COMING DOWN WITH SOMETHING
7. yes; m ∠ A = m ∠ X = 90°, and
m ∠ B = m ∠ C = m ∠ Y = m ∠ Z = 45°
8.2 Start Thinking
Sample answer: If the m ∠ A = 30°, you can conclude that m ∠ D = 60° because ADG is a right triangle. The parallel lines in the diagram are cut by two transversals creating congruent corresponding angles. ∠ A ≅ ∠ EBG ≅ ∠ FCG , so they all have a measure of 30°. In a similar way, you can conclude that ∠ D ≅ ∠ BEG ≅ ∠ CFG , so they all have a measure of 60°. 8.2 Warm Up 1. 56
2. 19
3. 122
1. 9
2. 7
3. 3.5
4. 14.6
5. 132°
6. 48°
8.2 Practice A
ABC ~ MLN ; ∠ A ≅ ∠ M ,
∠ B ≅ ∠ L , and ∠ C ≅ ∠ N , so ABC ~ MLN . 2. no; m ∠ F = 66° and m ∠ R = 95° 3. ∠ ADB ≅ ∠ E and ∠ A ≅ ∠ A, so
ABD ~ ACE .
4. ∠WXZ ≅ ∠ ZXY and ∠ W ≅ ∠ XZY , so
WXZ ~ ZXY .
5. a. yes; Because
ABC ~ EDC ,
∠ BAC ≅ ∠ CED. By the Alternate Interior
Angles Converse Theorem (Thm. 3.6), AB || DE. b. BE || AD, so ∠ EBD ≅ ∠ BDA by the
Alternate Interior Angles Theorem (Thm. 3.2). ∠ BCE ≅ ∠ DCA by the Vertical Angles Congruence Theorem (Thm. 2.6). So, ACD ~ ECB by the AA Similarity Theorem (Thm. 8.3).
A86
Geometry Answers
STATEMENTS
REASONS
1. ∠ Q ≅ ∠ T
1. Given
2. ∠ PRQ ≅
2. Vertical Angles Congruence Theorem (Thm. 2.6)
∠ SRT 3. PQR ~ STR
3. AA Similarity Theorem (Thm. 8.3)
4. ∠ P ≅ ∠ S
4. Corresponding parts of similar triangles are similar.
5. PQ || ST
5. Alternate Interior Angles Converse (Thm. 3.6)
4. 90
8.2 Cumulative Review Warm Up
1. yes;
8.
8.2 Practice B
WXY ~ STR; ∠W ≅ ∠ S , ∠ X ≅ ∠ T , and ∠ Y ≅ ∠ R , so WXY ~ STR.
1. yes;
2. no; m ∠ L = 32° and m ∠ JKM = 48° 3. ∠ C ≅ ∠ FDE and ∠ E ≅ ∠ E , so
ECG ~ EDF .
4. ∠ X ≅ ∠ Z and ∠ XWY ≅ ∠ ZYW , so
XWY ~ ZYW .
5. yes; m ∠ A = m ∠ X = 90° and
m ∠ B = m ∠ C = m ∠ Y = m ∠ Z = 45° 6. no; 75° + 105° = 180° 7. no; The corresponding angles may not be congruent
to each other. 8. 550 ft
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Answers 9.
STATEMENTS
REASONS
1. ∠ ABC and ∠ BDC are right angles.
1. Given
2. ∠ ABC ≅
2. Right Angles Congruence Theorem (Thm. 2.3)
∠ BDC 3. ∠ C ≅ ∠ C
STATEMENTS
10.
REASONS 1. Given
1. YZ ≅ YV XY ≅ WY
2. ∠V ≅ ∠ Z
∠W ≅ ∠ X 3. ∠ XYW ≅
4. AA Similarity Theorem (Thm. 8.3)
5. ∠ A ≅ ∠ CBD
5. Corresponding angles of similar triangles are congruent.
3. Vertical Angles Congruence Theorem (Thm. 2.6)
∠VYZ
3. Reflexive Property of Angle Congruence (Thm. 2.2)
4. ABC ~ BDC
2. Base Angles Theorem (Thm. 5.6)
4. Triangle Sum Theorem (Thm. 5.1)
4. m ∠ X + m ∠W + m ∠ XYW = 180°
m ∠V + m ∠ Z + m ∠VYZ = 180° 5. m ∠ X + m ∠W +
5. Transitive Property of Angle Congruence (Thm. 2.2)
m ∠ XYW = m ∠V + m ∠ Z + m ∠VYZ 6. m ∠ X + m ∠W +
6. Substitution Property of Equality
m ∠ XYW = m ∠V + m ∠ Z + m ∠ XYW
7. m ∠ X + m ∠W =
7. Subtraction Property of Equality
m ∠V + m ∠ Z 8. m ∠ X + m ∠ X =
8. Substitution Property of Equality
m∠Z + m∠Z
9. Simplify.
9. 2 m ∠ X = 2 m ∠ Z 10. m ∠ X = m ∠ Z
10. Division Property of Equality
11. XYW ≅ VYZ
11. AA Similarity Theorem (Thm. 8.3)
8.2 Enrichment and Extension 1. (6, 4), (6, − 4)
54 36 54 36 , , , − 13 13 13 13
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2. (0, 9), (0, − 9)
3.
4. (0, 4), (0, − 4)
5. (6, 9), (6, − 9)
6.
24 36 24 36 , , , − 13 13 13 13
Geometry Answers
A87
Answers 7.
4 x; Sample answer: Solve the proportion 3 a x = where PS = a. 8 3x a + x 3
8. STATEMENTS
1. Given
2. QS ⊥ PR
2. Definition of altitude
3. ∠ PSQ and ∠ QSR are right angles.
3. Definition of perpendicular
4. m ∠ PSQ =
4. Definition of right angle
∠ PQR 6. ∠ QSR ≅
∠ PQR 7. ∠ P ≅ ∠ P
8. ∠ R ≅ ∠ R
9. PSQ ~ PQR
1. Given
BG || CF
2. ∠ A ≅ ∠ E
2. Corresponding Angles Theorem (Thm. 3.1)
∠ EDF ≅ ∠ EHG
3. Vertical Angle Congruence Theorem (Thm. 2.6)
4. ∠ EDF ≅
4. Transitive Property of Angle Congruence (Thm. 2.2)
∠ AHB 5. ABH ~ EFD
5. AA Similarity Theorem (Thm. 8.3)
8.2 Puzzle Time
5. Definition of congruent angles 6. Definition of congruent angles 7. Reflexive Property of Angle Congruence (Thm. 2.2) 8. Reflexive Property of Angle Congruence (Thm. 2.2) 9. AA Similarity Theorem (Thm. 8.3) 10. AA Similarity Theorem (Thm. 8.3)
11. PSQ ~ QSR
11. Transitive Property of Congruency
Geometry Answers
1. AC || GE
∠ AHB
10. QSR ~ PQR
A88
REASONS
3. ∠ EHG ≅
m ∠ PQR = 90° 5. ∠ PSQ ≅
STATEMENTS
REASONS
1. ∠ PQR is a right angle QS is the altitude of PQR drawn from the right angle.
m ∠ QSR =
9.
A TOWEL 8.3 Start Thinking
Sample answer: The four-inch block measurements are x = 4 13 inches and y = 3 13 inches. The five-inch 5 inches and block measurements are x = 5 12
y = 4 16 inches. 8.3 Warm Up 1.
ABC ~ DEC or ABC ~ GEF
2.
FEG ~ CED or FEG ~ CBA
3. m ∠ ACB = 58°
4. m ∠ FEG = 32°
5. m ∠ ACE = 122°
6. AD || FG
8.3 Cumulative Review Warm Up 1. y =
1 x −3 5
1 9
2. y = − x +
3 2
4. y =
3. y = − x + 1
4 9
2 19 x+ 5 15
8.3 Practice A 1.
DEF
4.
15 21 18 3 = = = 35 49 42 7
2. 4
3. 9
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Answers 5.
6.
7.
AC EC = and ∠ C ≅ ∠ C , so BC DC ACE ~ BCD; SAS Similarity Theorem (Thm. 8.5)
7. x = 12, y = 16, z = 7 8.
STATEMENTS
1. Given
18 14 10 = = , so EFG ~ MNL; SSS 27 21 15 Similarity Theorem (Thm. 8.4)
AC AB = DF DE
2. m ∠ A = 43°
2. Given
VWZ ~ XYZ
3. ∠ A ≅ ∠ D
3. Transitive Property of Angle Congruence (Thm.2.2)
4. ABC ~ DEF
4. SAS Similarity Theorem (Thm. 8.5)
5. ∠ B ≅ ∠ E
5. Corresponding angles of similar triangles are congruent.
9. m ∠VWY = 54° 11. XY =
1.
m ∠ D = 43°
8. m ∠ VZY = 90° 10. m ∠WXY = 91.5°
185 ≈ 13.6
12. no; The lengths of the legs are not proportional. 13.
STATEMENTS 1.
PR TR = QR SR
2. ∠ R ≅ ∠ R
3. PRT ~ QRS
REASONS 1. Given
9.
2. Reflexive Property of Angle Congruence (Thm. 2.2)
4. Corresponding angles of similar triangles are congruent.
5. QS || PT
5. Corresponding Angles Converse (Thm. 3.5)
REASONS
1. LN = 2 x
1. Given
MN = 2 y NP = x NQ = y
3. SAS Similarity Theorem (Thm. 8.5)
4. ∠ RQS ≅ ∠ RPT
STATEMENTS
2.
3.
LN 2x = = 2 NP x MN 2y = = 2 NQ y LN MN = NP NQ
1. 8
2. 9
15 30 25 5 = = = 3. 12 24 20 4
5.
15 35 = and ∠ X ≅ ∠ X , so WXY ~ VXZ ; 27 63 SAS Similarity Theorem (Thm. 8.5) 10.5 18 12 = = , so LMN ~ RQP; SSS 7 12 8 Similarity Theorem (Thm. 8.4)
2. Ratio of corresponding sides
3. Transitive Property of Equality
4. ∠ LNM ≅ ∠ QNP
4. Vertical Angles Congruence Theorem (Thm. 2.6)
5. MLN ~ PQN
5. SAS Similarity Theorem (Thm. 8.5)
8.3 Practice B
4.
REASONS
8.3 Enrichment and Extension 1. a.
3 2
2. 33°
b.
9 4 3. about 15.4 ft
4. 26.7 ft
6. 15 units, 18 units
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Geometry Answers
A89
Answers 5. STATEMENTS
1. AH || CF
REASONS
8.4 Start Thinking
1. Given
Sample answer: ∠ ABE ≅ ∠ EBD ≅ ∠ BDC ≅ ∠ BCD , CB ≅ DB,
CA || FH
2. ∠ AHC ≅
∠ FCH 3. ∠ ACH ≅
∠ FHC 4. ∠ DKC ≅
∠ JKH 5. DKC ~ JKH
6.
DK KC = = JK KH DC JH
7. ∠CKB ≅
∠ HKG
2. Alternate Interior Angles Theorem (Thm. 3.2) 3. Alternate Interior Angles Theorem (Thm. 3.2) 4. Vertical Angles Congruence Theorem (Thm. 2.6) 5. AA Similarity Theorem (Thm. 8.3) 6. All sides of similar triangles are proportional.
CK KB 9. = = HK KG CB HG
9. All sides of similar triangles are proportional.
DK KB = 10 JK KG
10. Substitution
11. ∠ BKD ≅
11. Vertical Angles Congruence Theorem (Thm. 2.6)
8.3 Puzzle Time
2. x = −
3. x = 0, x = −15
4. x = −1, x = − 5
5. x = 5, x = 2
6. x = 2, x = 6
8.4 Cumulative Review Warm Up STATEMENTS
REASONS
1. AC ≅ AB,
1. Given
AD ≅ AE
2.
12. SAS Similarity Theorem (Thm. 8.5)
2. ∠ A ≅ ∠ A
2. Reflexive Property of Angle Congruence (Thm. 2.2)
3. ADB ≅ AEC
3. SAS Congruence Theorem (Thm. 5.5)
STATEMENTS
REASONS
1. MR ⊥ KP,
1. Given
KO ⊥ PM
2. ∠ RKM ≅ ∠ OMK
2. Given
3. ∠ MRK and ∠ KOM are right triangles.
3. Definition of perpendicular
4. ∠ MRK ≅ ∠ KOM
4. Right Angles Congruence Theorem (Thm. 2.3)
5. KM ≅ KM
5. Reflexive Property of Segment Congruence (Thm. 2.1)
6. RKM ≅ OMK
6. AAS Congruence Theorem (Thm.5.11)
AN ECHO
A90
Geometry Answers
9 2
1. x = − 5
7. Vertical Angles Congruence Theorem (Thm. 2.6) 8. AA Similarity Theorem (Thm. 8.3)
12 . BKJ ~ GKD
8.4 Warm Up
1.
8. CKB ~ HKG
∠GKJ
AD DC , = EB AE AD AC AE + ED AB + BC = ∴ = ∴ AE AB AE AB ED BC ED BC 1+ ; =1+ ∴ = AE AB AE AB ED DB . = By substituting DB for BC, you have AE AB
ADC ~ AEB ,
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Answers 8.4 Practice A
8.4 Enrichment and Extension
1. 6
2. 12
3. no
4. yes
5. 7.2
6. 2
7. 40
8. 9
9. 256 ft; If BE || CF , then by the Three Parallel
Lines Theorem (Thm. 8.8), By substitution,
AB BC . = DE EF
100 320 = and EF = 256 feet. 80 EF
8.4 Practice B 1. 37.5
2. 56
3. 45
4. 16
2 3
5. no; The Three Parallel Lines Theorem (Thm. 8.8)
proves that the parallel lines divide the transversals proportionally, so you cannot use it to prove that three lines are parallel. 6. yes; Let x equal the length of PN. You are given
enough information to write the equation x + ( 2 x − 9) = 45 to solve for x; LP = 27, PN = 18.
1. a = 22.8125, b = 15.625, c = 15, d = 8.33,
e = 4, f = 8 2. a = 9, b = 4, c = 3, d = 2 3. 22.1 in 4. a. 4
b. 3
c. 52 94 units
5. Because BE ⊥ AC and HG || AC , then
HG ⊥ BG by the Perpendicular Transversal Theorem (Thm. 3.11). Then ∠ GHA ≅ ∠ CAF by the Corresponding Angles Theorem (Thm. 3.1). Because ∠ AFC is also a right angle, AFC ~ HGB by the AA Similarity Theorem
AC BH . In BHG , = FC GB AH BH . Using substitution, = AE || HG , so, GE GB AC AH . Because we are given GE = FC , = FC GE it follows that AC = AH . (Thm. 8.3). So,
8.4 Puzzle Time 7.
STATEMENTS
REASONS
1. WY bisects ∠ XYZ .
1. Given
Cumulative Review
YW bisects ∠ XWZ .
2. ∠ XYV ≅ ∠ ZYV
∠ XWV ≅ ∠ ZWV
THE WHEELS BECAUSE THEY ARE ALWAYS TIRED
2. Definition of angle bisector
XV YZ 3. = XV VZ WX WZ = XV VZ
3. Triangle Angle Bisector Theorem (Thm. 8.9)
YZ 4. XY = • XV VZ WZ WX = • XV VZ
4. Multiplication Property of Equality
5. YZ ≅ WZ
1. no
2. yes
3. yes
4. no
5. no
6. yes
7. yes
8. yes
9. no
10. no
11. no
12. no
13. x = ± 6
14. x = ± 12
15. x = ± 3
16. x = ± 4
17. x = ± 8
18. x = ± 7
19. x = ± 11
20. x = ± 6
21. x = ± 13
22. x = ± 5
5. Given
23. x = ± 8
24. x = ± 7
6. XY = WX
6. Substitution
25. x = ± 3
26. x = ± 8
7. WXYZ is a kite.
7. Definition of a kite 27. x = ± 9
28. M − , 2
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5 2
Geometry Answers
A91
Answers 9 2
1 2
29. M − , −
31. M 5,
9 2
1 2
5 2
33. M − , −
30. M (9, − 4)
73. If WZ ≅ XZ , then ∠ ZWX ≅ ∠ ZXW ; Base
32. M ( 4, −1)
74. If XZ ≅ XY , then ∠ XZY ≅ ∠ Y ; Base Angles
34. M (9, 3)
75. If ∠ V ≅ ∠ WZV , then WV ≅ WZ ; Converse of
Angles Theorem (Thm. 5.6)
Theorem (Thm. 5.6)
Base Angles Theorem (Thm. 5.7)
3 5 2 2
76. If ZV ≅ ZY , then ∠V ≅ ∠ Y ; Base Angles
5 17 2 2
77. If ∠ ZWX ≅ ∠ ZXW , then ZW ≅ ZX ; Converse
7 2
78. If ∠ XZY ≅ ∠ Y , then ZX ≅ YX ; Converse of
11 7 , 2 2
36. M ,
37. M −
13 5 ,− 2 2
38. M − ,
39. M (5, 1)
40. M , − 6
35. M
Theorem (Thm. 5.6)
7 13 2 2
42. M ( − 5, − 3)
43.
181
44.
218
45.
61
46.
505
47.
97
48.
149
49.
197
50.
365
51.
509
52. 5
41. M ,
Base Angles Theorem (Thm. 5.7) 79. If ∠V ≅ ∠ Y , then ZV ≅ ZY ; Converse of Base
Angles Theorem (Thm. 5.7) 80. 16
54. 5 17
55. 2
56. 2
86. EF , FG , EG
87. ST , RS , RT
Chapter 9 9.1 Start Thinking
85
59. YW
60. YZ
61. XZ
62. ZV , ZW , ZX , ZY
63. ZW and ZY , or ZV and ZX
64. Sample answer: ZV and ZW 65. point M; 26
66. line s; 16
67. Mk ; 32
68. Mm; 10
69. line s; 84
70. point M; 24
71. Mm; 90
A92
Geometry Answers
83. 13
85. 5
c
58. XV
313
82. 10
84. 2
a
57.
81. 9
A = a 2 + b2
53. 11 5
29
of Base Angles Theorem (Thm. 5.7)
c b
b
a
A = c 2 ; Because the area of the original diagram must equal the area of the reassembled diagram, a 2 + b 2 = c 2 , which is a statement of the Pythagorean Theorem (Thm. 9.1). You have proved the theorem with your construction.
9.1 Warm Up 1. x = ± 5 3. x = ±
2. x = ± 2 114
221 6
4. x = ±
239
72. Mk ; 110
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