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LESSON
3.3
Answers for the lesson “Prove Lines
are Parallel”
Skill Practice
1. Sample:
12. no
13. yes; Corresponding Angles
n
Converse
12
3 4
5 6
7 8
14. no
15. yes; Alternate Exterior Angles
m
Converse
16. Sample answer:
Ž1 and Ž8, Ž2 and Ž7
4 1
3 2
50
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2. Given two lines cut by a
transversal, alternate interior
angles are congruent if and only
if the lines are parallel; given two
lines cut by a transversal,
alternate exterior angles are
congruent if and only if the lines
are parallel; given two lines cut
by a transversal, consecutive
interior angles are supplementary
if and only if the lines are parallel.
3. 40
4. 60
5. 15
6. 90
7. 60
8. 20
9. The student believes that x 5 y
but there is no indication that they
are equal.
8 5
7 6
mŽ3 5 508, Vertical Angles
Congruence Theorem;
mŽ4 5 1308, Linear Pair
Postulate;
mŽ2 5 1308, Vertical Angles
Congruence Theorem;
mŽ8 5 1308, Alternate Interior
Angles Theorem;
mŽ6 5 1308, Vertical Angles
Congruence Theorem;
mŽ5 5 508, Linear Pair
Postulate;
mŽ7 5 508, Vertical Angles
Congruence Theorem
10. yes; Alternate Interior Angles
Converse
11. yes; Alternate Exterior Angles
Converse
Geometry
Answer Transparencies for Checking Homework
68
17. a. mŽDCG 5 1158,
24. 1 angle. Sample answer: Using
mŽCGH 5 658
the Vertical Angles Congruence
Theorem, the Linear Pair
Postulate, and the Alternate
Interior Angles Theorem the other
angle measures can be found.
b. They are consecutive
interior angles and they are
supplementary.
c. yes; Consecutive Interior
25. Sample answer: Ž1 > Ž4
Angles Converse
18. a. Sample:
p
1
2
q
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b. Given: Ž1 and Ž2 are
supplementary, Prove: p i q
19. yes; Consecutive Interior Angles
Converse
20. yes; Alternate Exterior Angles
Converse
21. no
22. The student assumed the
congruent angles were
alternate interior angles
‹]›
‹]›
between AD
and BC
. By
the Alternate Interior
‹]› ‹]›
Angles Converse; AB
i DC .
23. D
therefore Ž4 and Ž7 are
supplementary. Lines j and k are
parallel by the Consecutive
Interior Angles Converse.
]› ]› ]›
26. EA i HC ; EB is not parallel
]›, ŽGHC >ŽHEA, ŽGHD
to HD
is not congruent to ŽHEB.
27. a. 1 line
b. an infinite number of lines
c. 1 plane
28. a. 54
b. 47.5
c. No, Sample answer: For p to be
parallel to q, x 5 54, then
y 5 63 because of the linear
pair formed, but in order for
r and s to be parallel, y must
equal 47.5.
Problem Solving
29. Alternate Interior Angles
Converse
30. Corresponding Angles Converse
Geometry
Answer Transparencies for Checking Homework
69
31. 3. Substitution
35. Statements (Reasons)
4. Definition of supplementary
angles
5. Consecutive Interior Angles
Converse
32. Alternate Exterior Angles
Converse
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33. Yes. Sample answer: E 20th is
parallel to E 19th by the
Corresponding Angles Converse.
E 19th is parallel to E 18th by
the Alternate Exterior Angles
Converse. E 18th is parallel to
E 17th by the Alternate Interior
Angles Converse. They are all
parallel by the Transitive Property
of Parallel Lines.
34. Statements (Reasons)
1. Ž1 > Ž2, Ž3 > Ž4 (Given)
2. Ž2 > Ž3
(Vertical Angles
Congruence Theorem)
3. Ž1 > Ž4
(Transitive Property
of Angle Congruence)
4. }
AB i }
CD
(Alternate Interior
Angles Converse)
1. a i b, Ž2 > Ž3
(Given)
2. Ž2 and Ž4 are supplementary.
(Consecutive Interior
Angles Theorem)
3. Ž3 and Ž4 are supplementary.
(Substitution)
4. c i d
(Consecutive Interior
Angles Converse)
36. Statements (Reasons)
1. Ž2 > Ž7
(Given)
2. Ž7 > Ž6
(Vertical Angles
Congruence Theorem)
3. Ž2 > Ž6
(Transitive
Property of Congruence)
4. m i n
(Corresponding
Angles Converse)
37. You are given that Ž3 and Ž5 are
supplementary. By the Linear Pair
Postulate, Ž5 and Ž6 are also
supplementary. So Ž3 > Ž6 by
the Congruent Supplements
Theorem. By the Alternate
Interior Angles Converse, m i n.
Geometry
Answer Transparencies for Checking Homework
70
38. a.
1
p
q
2
4
r
40–44. Sample answers are given.
40. Consecutive Interior Angles
t
3
Converse
41. Corresponding Angles Converse
42. Corresponding Angles Converse
b. Given: p i q and q i r,
43. Vertical Angles Congruence
Prove: p i r
Theorem followed by the
Corresponding Angles Converse
c. Statements (Reasons)
1. p i q andq i r
(Given)
2. Ž1 > Ž2
(Alternate
Interior Angles Theorem)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
3. Ž2 > Ž3 (Vertical Angles
Congruence Theorem)
4. Ž3 > Ž4
(Alternate
Interior Angles Theorem)
5. Ž1 > Ž4
(Transitive Property
of Angle Congruence)
6. p i r
(Alternate Interior
Angles Converse)
44. Consecutive Interior Angles
Converse
45. a.
Q
A
1
C
2
3
4
P
D
B
n
b. If two parallel lines are cut by
a transversal, the angle
bisectors of alternate interior
angle pairs are parallel.
39. a. Sample answer: Corresponding
Angles Converse Theorem
b. Slide the triangle along a fixed
horizontal line and use the
edge that forms the 908 angle
to draw vertical lines.
Geometry
Answer Transparencies for Checking Homework
71
Mixed Review of
Problem Solving
45. b. (cont.)
Statements (Reasons)
1. * i n
(Given)
2. ŽAQP > ŽBPQ
(Alternate Interior
Angles Theorem)
3. mŽ1 1 Ž2 5 mŽAQP,
mŽ4 1 Ž3 5 mŽBPQ
(Angle Addition Postulate)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
4. mŽ1 5 mŽ2,
mŽ3 5 mŽ4
(Definition of
angle bisector)
1. a. Sample answer: q and p,
k and m
b. Sample answer: q and m
c. Sample answer: n and m,
n and k
2. a. Ž2: supplementary,
Ž3: supplementary,
Ž4: vertical,
Ž5: corresponding,
Ž6: supplementary,
Ž7: alternate exterior,
Ž8: supplementary
5. mŽ2 1 mŽ2 5 mŽAQP,
mŽ3 1 mŽ3 5 mŽBPQ
(Subtitution)
3. 538; Alternate Exterior Angles
6. 2mŽ2 5 2mŽ3 (Transitive
Property of Equality)
4. yes; Alternate Interior Angles
7. mŽ2 5 mŽ3
(Division
Property of Equality)
5. a. 11
8. Ž2 > Ž3 (Definition of
Congruent Angles)
]› i PD
]›
9. QC
(Alternate
Interior Angles Converse)
b. Ž2, Ž6, Ž8
Theorem
Converse
b. 238; Transitive Property
of Parallel Lines and
Alternate Interior Angles
Theorem
Geometry
Answer Transparencies for Checking Homework
72
6. 1508;
1
5
0
7. 92, supplementary to 888;
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
x 5 116, c i d by the Alternate
Interior Angles Converse followed
by the Consecutive Interior
Angles Theorem.
Geometry
Answer Transparencies for Checking Homework
73