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@##$
Chapter 3
continued
17. One line can be drawn through A and perpendicular
to @##$
BC.
36. Answers will vary.
37. A; The horizontal bars are parallel.
m
38. Answers will vary.
A
39. The adjacent interior angles are supplementary, so the
measure of the other two interior angles must be 908.
C
40. True; the plane containing the floor of the treehouse is
B
parallel to the ground.
18. Ž 5 and Ž 1 are corresponding angles.
19. Ž 11 and Ž 13 are consecutive interior angles.
20. Ž 6 and Ž 13 are consecutive interior angles.
21. Ž 10 and Ž 15 are alternate exterior angles.
41. False; the lines containing the railings of the staircase
intersect the ground, so they are not skew to the ground.
42. True; the lines containing the balusters are perpendicular
to the plane containing the floor.
l
43. Sample answer:
22. Ž 2 and Ž 11 are alternate interior angles.
m
23. Ž 8 and Ž 4 are corresponding angles.
P
n
24. If two lines are parallel, then they are always coplanar.
P
l
m
44. Sample answer:
A
25. If two lines are not coplanar, then they never intersect.
l
m
B
Mixed Review for TAKS
l
P
45. B;
26(4 2 x) 1 3(2x 1 5) 5 224 1 6x 1 6x 1 15
5 12x 2 9
26. If three lines intersect at one point, then they are
sometimes coplanar.
Œr
p
H
m
or
P
X
l
72 1 b2 5 252
49 1 b2 5 625
b2 5 576
27. If two lines are skew to a third line, then they are
sometimes skew to each other.
N
M
O
N
O
M
Q
R
L
P
The perimeter of the square is 192 inches.
R
L
P
Lesson 3.2
28. B; Ž RPQ and Ž PRS are alternate interior angles.
Investigating Geometry Activity 3.2 (p. 153)
29. Ž BCG, Ž CFJ, and Ž GJH are corresponding angles.
1–2. Answers for the table will vary. So, Ž AGE > Ž BGH
30. Ž BCG and Ž HJC are consecutive interior angles.
31. Ž FCJ, Ž HJC, and Ž DFC are alternate interior angles.
32. Ž FCA and Ž GJH are alternate exterior angles.
33. a. mŽ 1 5 808; mŽ 2 5 808
b. mŽ 3 5 708; mŽ 4 5 708
c. If parallel lines are cut by a transversal, then the
alternate exterior angles are congruent.
> Ž CHG > Ž DHF, Ž EGB > Ž AGH
> Ž GHD > Ž CHF.
3. a. When two parallel lines are cut by a transversal,
corresponding angles are congruent.
b. When two parallel lines are cut by a transversal,
alternate interior angles are congruent.
4. When two parallel lines are cut by a transversal, the
consecutive interior angles are supplementary.
Problem Solving
When mŽ AGH 5 708, then mŽ CHG 5 1108.
34. The platform is parallel to the ground.
When mŽ BGH 5 1308, then mŽ GHD 5 508.
35. The arm is skew to a telephone pole.
56
The other leg of the triangle is 24 inches. So, the side
length s of the square is 2(24) 5 48 inches.
P 5 4s 5 4(48) 5 192
Q
or
b 5 24
Geometry
Worked-Out Solution Key
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
a2 1 b2 5 c2
n
t
46. H;
Chapter 3,
continued
3.2 Guided Practice (pp. 155–156)
1. mŽ 4 5 1058; Vertical Angles Congruence Theorem;
mŽ 5 5 1058; Corresponding Angles Postulate;
mŽ 8 5 1058; Alternate Exterior Angles Theorem
17. mŽ 1 5 1508, by the Corresponding Angles Postulate.
mŽ 2 5 1508, by the Alternate Exterior Angles Theorem.
18. mŽ 1 5 1408, by the Alternate Interior Angles Theorem.
mŽ2 5 408, by the Consecutive Interior
Angles Theorem.
2. mŽ 7 1 mŽ 8 5 1808
mŽ 3 5 mŽ 7
19. mŽ 1 5 1228, by the Alternate Interior Angles Theorem.
mŽ 3 1 mŽ 8 5 1808
mŽ 2 5 588, by the Consecutive Interior Angles
Theorem.
688 1 (2x 1 4)8 5 1808
2x 5 108
20. The student cannot conclude that Ž 9 > Ž 10 by the
x 5 54
The value of x is 54.
3. Yes. You could still prove the theorem because the
congruence of Ž 3 and Ž 2 is not dependent on the
congruence of Ž 1 and Ž 3.
4. Because the sun’s rays are parallel, Ž 1 and Ž 2 are
alternate interior angles. By the Alternate Interior Angles
Theorem, Ž 1 > Ž 2. By the definition of congruent
angles, mŽ 1 5 mŽ 2 5 418.
3.2 Exercises (pp. 157–160)
Skill Practice
1. Sample answer:
1
2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
16. Consecutive Interior Angles Theorem
2. When two parallel lines are cut by a transversal, vertical
angles, corresponding angles, alternate interior angles,
and alternate exterior angles are congruent and adjacent
angles, exterior angles on the same side of the transversal
and consecutive interior angles are supplementary.
3. C; Ž 4 and Ž 1 are corresponding angles,
so mŽ 4 5 mŽ 1.
4. If mŽ 4 5 658, then mŽ 1 5 658, by the Vertical Angles
Congruence Theorem.
5. If mŽ 7 5 1108, then mŽ 2 5 1108, by the Alternate
Exterior Angles Theorem.
6. If mŽ 5 5 718, then mŽ 4 5 718, by the Alternate
Interior Angles Theorem.
7. If mŽ 3 5 1178, then mŽ 5 5 638, by the Consecutive
Interior Angles Theorem.
8. If mŽ 8 5 548, then mŽ 1 5 548, by the Alternate
Exterior Angles Theorem.
9. Corresponding Angles Postulate
10. Alternate Interior Angles Theorem
11. Alternate Interior Angles Theorem
12. Consecutive Interior Angles Theorem
13. Alternate Exterior Angles Theorem
Corresponding Angles Postulate because the lines are not
known to be parallel. There is not enough information
given for the student to make any conclusion about the
angles.
21. Sample answer:You can show that Ž 1 > Ž 4 by the
Alternate Exterior Angles Theorem. You can show that
Ž 1 > Ž 3 by the Corresponding Angles Postulate and
Ž 3 > Ž 4 by the Vertical Angles Congruence Theorem.
So, Ž >Ž 4 by the Transitive Property of Angle
Congruence.
22. Ž 1 and the 808 angle are supplementary by the
Consecutive Interior Angles Theorem, so mŽ 1 5 1008.
Ž 1 and Ž 2 are supplementary by the Consecutive
Interior Angles Theorem, so mŽ 2 5 808. Ž 3 and the
808 angle are supplementary by the Consecutive Interior
Angles Theorem, so mŽ 3 5 1008.
23. Ž 1 and the 908 angle are supplementary by the
Consecutive Interior Angles Theorem, so mŽ 1 5 908.
Ž 2 is congruent to the 1158 angle by the Corresponding
Angles Postulate, so mŽ 2 5 1158. Ž 2 and Ž 3 are
supplementary by the Consecutive Interior Angles
Theorem, so mŽ 3 5 658.
24. Ž 2 is congruent to the 1338 angle by the Alternate
Interior Angles Theorem, so mŽ 2 5 1338. Ž 3 and the
1338 angle are supplementary by the Consecutive Interior
Angles Theorem, so mŽ 3 5 478. Ž 1 > Ž 3 by the
Alternate Interior Angles Theorem, so mŽ 1 5 478.
@##$, then Ž BAC > Ž DCA
25. Sample answer: If @##$
AB i DC
and Ž CDB > Ž ABD.
@##$ then Ž BAD and Ž ABC
26. Sample answer: If @##$
AD i BC,
are supplementary and Ž ADC and ŽBCD
are supplementary.
27. Using the Alternate Interior Angles Theorem, x 5 45.
Using the Corresponding Angles Postulate, y 5 85.
28. Using the Consecutive Interior Angles Theorem:
3y8 1 6y8 5 1808
9y 5 180
y 5 20
2x8 1 908 5 1808
2x 5 90
x 5 45
14. Corresponding Angles Postulate
15. Alternate Exterior Angles Theorem
Geometry
Worked-Out Solution Key
57
Chapter 3,
continued
29. Using the Corresponding Angles Postulate, x 5 65.
35. Using the Consecutive Interior Angles Theorem:
(2x 2 y)8 1 608 5 1808
Using the Angle Addition Postulate:
x8 1 y8 1 558 5 1808
2x 2 y 5 120 l y 5 2x 2 120
65 1 y 1 55 5 180
(2x 1 y)8 1 408 5 1808
y 5 60
2x 1 y 5 140
30. Using the Corresponding Angles Postulate:
2x8 1 (2x 2 120)8 5 1408
3x8 5 608
4x 5 260
x 5 20
x 5 65
Using the Consecutive Interior Angles Theorem:
y 5 2(65) 2 120 5 10
(5y 2 5)8 1 1358 5 1808
So, the value of x is 65 and the value of y is 10.
5y 5 50
36. Using the Consecutive Interior Angles Theorem:
y 5 10
1508 1 (5x 2 y)8 5 1808
31. Using the Alternate Interior Angles Theorem:
5x 2 y 5 30 l y 5 5x 2 30
4x8 5 528
1308 1 (5x 1 y)8 5 1808
x 5 13
5x 1 y 5 50
Using the Consecutive Interior Angles Theorem:
5x8 1 (5x 2 30)8 5 508
F 4x8 1 (3y 1 2)8 G 1 908 5 1808
10x 5 80
4(13) 1 3y 1 2 1 90 5 180
x58
3y 5 36
y 5 5(8) 2 30 5 10
y 5 12
The value of x is 8 and the value of y is 10.
32. Using the Consecutive Interior Angles Theorem:
5x8 1 (14x 2 10)8 5 1808
Problem Solving
19x 5 190
Statements
Reasons
2y8 1 (14x 2 10)8 5 1808
1. p i q
1. Given
2y 1 14(10) 2 10 5 180
2. Ž1 > Ž3
2. Corresponding
Angles Postulate
3. Ž3 > Ž2
3. Vertical Angles
Congruence Postulate
4. Ž1 > Ž2
4. Transitive Property
of Angle Congruence
2y 5 50
y 5 25
33. B; Using the Alternate Exterior Angles Theorem:
mŽ 1 5 1108
Using the Linear Pair Postulate:
( y 2 5)8 1 mŽ1 5 1808
38. a. Ž 4 > Ž 1 by the Vertical Angles Congruence
( y 2 5)8 1 1108 5 1808
y 5 75
34. Sample answer:
M
P
Theorem. Ž 5 > Ž 1 by the Corresponding Angles
Postulate. Ž 8 > Ž 1 by the Alternate Exterior
Angles Theorem.
b. Ž 6 and Ž 8 are a linear pair, so they are
N
Q
Ž MNQ and Ž PQN, and Ž PQN and Ž QPM are
both supplementary by the Consecutive Interior
Angles Theorem. ŽMNQ > ŽQPM by the Congruent
Supplements Theorem. Ž NMP and Ž QPM, and Ž QPM
and Ž PQN are both supplementary by the Consecutive
Interior Angles Theorem. Ž NMP > Ž PQN by the
Congruent Supplements Theorem. So, Ž MNQ > Ž QPM
and Ž NMP > Ž PQN.
58
Geometry
Worked-Out Solution Key
supplementary and mŽ 6 1 mŽ 8 5 1808.
mŽ 8 5 mŽ 1 5 1108. So, mŽ 6 1 1108 5 1808,
or mŽ 6 5 708.
39. a. The following pairs of angles are always congruent:
Ž 1 and Ž 5, and Ž 2 and Ž 6.
The following pairs of angles are always
supplementary: Ž 1 and Ž 2, Ž 1 and Ž 6, Ž 2 and Ž 5,
and Ž 5 and Ž 6.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
37.
x 5 10
Chapter 3,
continued
b. Because the bars are parallel, the corresponding angles
between the bars and the foot are congruent. Because
the body and the foot are parallel, the bars act as
transversals, and so the alternate interior angles are
congruent. (See diagram.) This forces the foot to stay
parallel with the floor.
43. Ž 4>Ž 2 by the Alternate Interior Angles Theorem. Ž 2
> Ž 3 by the definition of angle bisector. Ž 5 > Ž 1 by
the Corresponding Angles Postulate. Ž 4 > Ž 5 is given,
so Ž 1 > Ž 2 > Ž 3 > Ž 4 > Ž 5. Because
mŽ 1 1 mŽ2 1 mŽ 3 5 1808, the measure of each angle
is 608. So, mŽ 1 5 608.
Mixed Review for TAKS
44. B;
Let * be the length and let w be the width of the original
rectangle.
40. a. Using the Alternate Interior Angles Theorem,
Pnew 5 2(2*) 1 2(2w)
5 2(2* 1 2w)
mŽ 2 5 mŽ 5 708. Ž 2 and Ž 3 are a linear pair,
so 708 1 mŽ3 5 1808, or mŽ 3 5 1108.
5 2(20)
5 40
b. Ž ABC is a straight angle because Ž 2 and Ž 3 are
supplementary, so the sum of the angle measures
is 1808.
c. If mŽ 1 is 608, Ž ABC will still be a straight angle
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
41.
42.
because Ž 2 and Ž 3 will still be supplementary. The
opening of the box will be more steep because the
measure of Ž 1 is smaller, so the slope of the line
becomes greater.
Statements
Reasons
1. n i p
1. Given
2. Ž1 > Ž3
2. Alternate Interior Angles
Theorem
3. mŽ 1 5 mŽ 3
3. Definition of congruent
angles
4. Ž 2 and Ž 3 form
a linear pair.
4. Definition of linear pair
5. mŽ 2 1 mŽ 3 5 1808
5. Linear Pair Postulate
6. mŽ 1 1 mŽ 2 5 1808
6. Substitution Property of
Equality
7.Ž 1 and Ž 2 are
supplementary.
7. Definition of
supplementary angles
The perimeter of the new rectangle is 40 feet.
45. F;
18x3y24
18
15
6x2z2
5y
6
5
}
5 }x3 2 1y24 2 1z2 5 }x2y25z2 5 }
5
22
15xyz
46. C;
The expression x 1 0.18x represents the total cost of a
meal including an 18% tip, where x is the cost of the food
in dollars.
Quiz 3.1–3.2 (p. 160)
Statements
Reasons
1. t >r, r i s
1. Given
2. Ž1 is a right angle.
2. Definition of perpendicular
lines
3. Ž1 > Ž 2
3. Corresponding Angles
Postulate
4. Ž 2 is a right angle.
4. Definition of congruent
angles
5. t > s
5. Definition of perpendicular
lines
1. Ž 2 and Ž 6 are corresponding angles.
2. Ž3 and Ž5 are consecutive interior angles.
3. Ž3 and Ž6 are alternate interior angles.
4. Ž2 and Ž7 are alternate exterior angles.
5. Using the Alternate Interior Angles Theorem:
2x8 5 1288
x 5 64
6. Using the Alternate Exterior Angles Theorem:
(2x 1 1)8 5 1518
2x8 5 1508
x 5 75
7. Using the Consecutive Interior Angles Theorem:
(7x 1 24)8 1 728 5 180
7x 5 84
x 5 12
Lesson 3.3
3.3 Guided Practice (pp. 161–164)
1. Yes. The measure of the angle corresponding to the
758 angle is 758 because it forms a linear pair with the
1058 angle. Then, corresponding angles are congruent
by defintion of congruent angles. So m i n by the
Corresponding Angles Converse.
2. Postulate 16 exchanges the hypothesis and conclusion
of Postulate 15, so Postulate 16 is the converse of
Postulate 15.
Geometry
Worked-Out Solution Key
59
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