Download Chapter 3-Parallel and Perpendicular Lines

Chapter 3-Parallel and Perpendicular Lines
2. Which statement is true?
Multiple Choice
Identify the choice that best completes the statement or
answers the question.
1. Which angles are corresponding angles?
a.
b.
c.
d. none of these
3. Which is a correct two-column proof?
Given:
Prove:
and
n
p
d
l
b
c
h
j
a.
b.
are supplementary.
k
m
a.
b.
c.
d.
are same-side angles.
are same-side angles.
are alternate interior angles.
are alternate interior angles.
c.
d. none of these
4. Line r is parallel to line t. Find m 5. The diagram is not to scale.
r
7
135°
1
3
t
4
a. 45
b. 35
c. 135
2
5
6
d. 145
5. Find the value of the variable if
1
2
3
l
4
5
6
7
a. 1
b. 2
c. 3
8
m
d. –2
6. Find the values of x and y. The diagram is not to scale.
and
The diagram is not to scale.
(x – 3)°
41°
(y + 8)°
74°
a. x = 77, y = 59
b. x = 77, y = 57
c. x = 57, y = 77
d. x = 41, y = 57
7. Complete the statement. If a transversal intersects two parallel lines, then ____.
a. corresponding angles are supplementary b. same-side interior angles are complementary
angles are congruent d. none of these
c. alternate interior
8. Complete the statement. If a transversal intersects two parallel lines, then ____ angles are supplementary.
a. acute b. alternate interior c. same-side interior d. corresponding
9. Find
The diagram is not to scale.
Q
R
70°
50°
a. 60
b. 120
10.
Which is a correct two-column proof?
Given:
Prove:
c. 110
and
are supplementary.
C
P Q
R S
T U
V W
a.
d. 70
B
Y
b.
c.
d. none of these
11. Which lines, if any, can you conclude are parallel
given that
? Justify your
conclusion with a theorem or postulate.
3 4
5
1 2
6
g
p
1
j
2
h
k
a.
, by the Converse of the Same-Side Interior
Angles Theorem b.
, by the Converse of
the Alternate Interior Angles Theorem c.
,
by the Converse of the Alternate Interior Angles
Theorem d.
, by the Converse of the
Same-Side Interior Angles Theorem
12.
q
. Find the value of x for p
to be parallel to q. The diagram is not to scale.
a. 114
13. If
b. 126
and
c. 120
d. 20
, what is
b
1 2
3 4
a
5 6
7 8
c
a. 90
b. 106
c. 74
d. not enough information
72°
14. Find the value of k. The diagram is not to scale.
62°
105°
a. 33
k°
45°
a. 17
b. 73
c. 118
d. 107
b. 162
x°
c. 147
d. 75
19. The folding chair has different settings that change
the angles formed by its parts. Suppose
is 26
and
is 70. Find
. The diagram is not to
scale.
15. Find the values of x, y, and z. The diagram is not to
scale.
1
2
3
38°
19°
a. 96
56°
x°
z°
b. 106
c. 116
d. 86
y°
20. Find the value of the variable. The diagram is not to
scale.
a.
b.
c.
d.
114°
16. Classify the triangle by its sides. The diagram is not
to scale.
a. 66
9
b. 19
c. 29
d. 43
21. Find the value of x. The diagram is not to scale.
9
Given:
9
a. straight b. scalene
d. equilateral
47°
x°
c. isosceles
17. Classify ABC by its angles, when m A = 32,
m B = 85, and m C = 63.
a. right b. straight c. obtuse d. acute
18. Find the value of x. The diagram is not to scale.
,
,
a.
S
b.
c.
R
a. 5
T
b. 24
c. 20
U
d. 40
d.
22. What is a correct name for the polygon?
A
B
E
24. Classify the polygon by its sides.
C
D
a. EDCAB
b. ABCDA
c. CDEAB
d. BAEAB
23. Which figure is a convex polygon?
a. triangle
d. octagon
b. hexagon
c. pentagon
25. Find the sum of the measures of the angles of the
figure.
a. 540
b. 180
c. 360
d. 900
26. How many sides does a regular polygon have if
each exterior angle measures 20?
a. 17 sides b. 20 sides c. 21 sides d. 18 sides
27. Find the missing angle measures. The diagram is
not to scale.
125°
x°
124° y°
3
34. Graph y =  x – 1.
4
65°
a. x = 124, y = 125 b. x = 56, y = 114
114, y = 56 d. x = 56, y = 124
c. x =
28. The sum of the measures of two exterior angles of a
triangle is 255. What is the measure of the third
exterior angle?
a. 75 b. 115 c. 105 d. 95
29. The Polygon Angle-Sum Theorem states: The sum
of the measures of the angles of an n-gon is ____.
a.
b.
c.
d.
30. Complete this statement. The sum of the measures
of the exterior angles of an n-gon, one at each
vertex, is ____.
a. (n – 2)180
b. 360
c.
d. 180n
31. Complete this statement. A polygon whose sides all
have the same length is said to be ____.
a. regular b. equilateral c. equiangular
d. convex
32. Find
. The diagram is not to scale.
96°
118°
115°
104°
a. 107
b. 117
A
c. 63
33. A nonregular hexagon has five exterior angle
measures of 55, 60, 69, 57, and 57. What is the
measure of the interior angle adjacent to the sixth
exterior angle?
a. 128 b. 118 c. 62 d. 108
d. 73
a.
c.
y
–6
–4
6
6
4
4
2
2
–2
2
4
6
–4
–2
–2
–4
–4
–6
–6
d.
y
–4
–6
x
–2
b.
–6
y
6
4
4
2
2
2
4
6
–6
x
–4
–2
–2
–2
–4
–4
–6
–6
35. Graph
4
6
x
2
4
6
x
y
6
–2
2
.
a.
c.
y
y
8
8
6
6
4
4
2
2
–8 –6 –4 –2
–2
2
4
6
8
–8 –6 –4 –2
–2
x
–4
–4
–6
–6
–8
–8
b.
d.
y
8
6
6
4
4
2
2
2
4
6
8
x
4
6
8
x
2
4
6
8
x
y
8
–8 –6 –4 –2
–2
2
–8 –6 –4 –2
–2
–4
–4
–6
–6
–8
–8
36. Write an equation in point-slope form of the line
through point J(–5, 6) with slope –4.
a.
b.
c.
d.
37. Write an equation in point-slope form, y – y1 = m(x
– x1), of the line through points (4, –4) and (1, 2)
Use (4, –4) as the point (x1, y1).
a. (y – 4) = –2(x + 4) b. (y – 4) = 2(x + 4) c. (y
+ 4) = 2(x – 4) d. (y + 4) = –2(x – 4)
38. Write an equation for the horizontal line that
contains point E(–3, –1).
a. x = –1 b. x = –3 c. y = –1 d. y = –3
39. Graph the line that goes through point (–5, 5) with
1
slope .
5
a.
c.
y
–6
–4
y
6
6
4
4
2
2
–2
2
4
6
–6
x
–4
–2
–2
–2
–4
–4
–6
–6
d.
b.
x
2
4
6
x
4
4
2
2
–2
6
6
6
–4
4
y
y
–6
2
2
–2
–4
–6
4
6
x
–6
–4
–2
–2
–4
–6
40. Write an equation in slope-intercept form of the
line through point P(–10, 1) with slope –5.
a. y = –5x – 49 b. y – 1 = –5(x + 10) c. y – 10 =
–5(x + 1) d. y = –5x + 1
41. Write an equation in slope-intercept form of the
line through points S(–10, –3) and T(–1, 1).
4
13
4
13
4
 x+
 x
a.
b. y = x –
c.
9
9
9
9
9
13
4
13
–
d. y = x +
9
9
9
42. Is the line through points P(0, 5) and Q(–1, 8) parallel to the line through points R(3, 3) and S(5, –1)? Explain.
a. No, the lines have unequal slopes. b. Yes; the lines are both vertical. c. Yes; the lines have equal slopes.
d. No, one line has slope, the other has no slope.
43. Which two lines are parallel?
I.
II.
III.
a. I and II
b. I and III
c. II and III
d. No two of the lines are parallel.
44. Write an equation for the line parallel to y = –7x + 15 that contains P(9, –6).
a. x + 6 = 7(y – 9) b. y + 6 = 7(x – 9) c. y – 6 = –7(x – 9) d. y + 6 = –7(x – 9)
45. Is the line through points P(10, –9) and Q(12, –15) perpendicular to the line through points R(–8, 9) and S(–14, 7)?
Explain.
a. Yes; their slopes are equal. b. No, their slopes are not opposite reciprocals. c. Yes; their slopes have product
–1 d. No; their slopes are not equal
46. Write an equation for the line perpendicular to y =
47. What must be true about the slopes of two
2x – 5 that contains (–9, 6).
perpendicular lines, neither of which is vertical?
a. y – 6 = 2(x + 9) b. x – 6 = 2(y + 9) c. y – 9 =
a. The slopes are equal. b. The slopes have
product 1. c. The slopes have product –1.
1
1
 (x + 6) d. y – 6 =  (x + 9)
d. One of the slopes must be 0.
2
2
48. Are the lines y = –x – 2 and 4x + 4y = 16 perpendicular? Explain.
a. Yes; their slopes have product –1. b. No; their slopes are not opposite reciprocals.
equal. d. No; their slopes are not equal
c. Yes; their slopes are
49. Give the slope-intercept form of the equation of the line that is perpendicular to
7x + 3y = 18 and contains P(6, 8).
3
3
18
3
38
3
a. y – 6 = (x – 8) b. y = x 
c. y = x +
d. y – 8 = (x – 6)
7
7
7
7
7
7
Short Answer
50. Give the missing reasons in this proof of the
Alternate Interior Angles Theorem.
Given:
Prove:
51. State the missing reasons in this proof.
Given:
Prove:
q
1 2
3 4
7
5 6
8
p
r
52. Find the measure of each interior and exterior
angle. The diagram is not to scale.
5
4
53. Find the measures of an interior angle and an
exterior angle of a regular polygon with 6 sides.
54. Write the equation 10x + 5y = 5 in slope-intercept
form. Then graph the line.
122 o
6
1
2
9
8
3
7
Essay
55. Write a paragraph proof of this theorem: In a plane,
if two lines are perpendicular to the same line, then
they are parallel to each other.
Given:
Prove:
1
2
3
l
4
s
1
2
3
4
r
5
6
7
5
6
7
8
8
m
t
57. Find the values of the variables. Show your work
and explain your steps. The diagram is not to scale.
o
31
56. Write a two-column proof.
Given:
Prove:
are supplementary.
x
w
v
y
o
68
z
Other
58. Given
, what can you conclude about
the lines l, m, and n? Explain.
n
1
l
2
l
m
59. Is each figure a polygon? If yes, describe it as
concave or convex and classify it by its sides. If
not, tell why.
a.
b.
c.
60. Explain how to tell whether a polygon is convex.
Chapter 3-Parallel and Perpendicular Lines
Answer Section
MULTIPLE CHOICE
1. ANS:
STA:
2. ANS:
STA:
3. ANS:
STA:
4. ANS:
STA:
5. ANS:
STA:
6. ANS:
STA:
7. ANS:
STA:
8. ANS:
STA:
9. ANS:
STA:
10. ANS:
STA:
11. ANS:
STA:
12. ANS:
STA:
13. ANS:
STA:
14. ANS:
STA:
15. ANS:
STA:
16. ANS:
STA:
17. ANS:
STA:
18. ANS:
STA:
19. ANS:
STA:
20. ANS:
STA:
21. ANS:
STA:
A
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
D
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
A
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
C
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
B
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
B
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
C
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
C
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
A
REF: 3-1 Properties of Parallel Lines
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
A
REF: 3-2 Proving Lines Parallel
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
A
REF: 3-2 Proving Lines Parallel
CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
D
REF: 3-3 Parallel and Perpendicular Lines
CA GEOM 7.0
A
REF: 3-3 Parallel and Perpendicular Lines
CA GEOM 7.0
B
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
A
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
A
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
CA GEOM 12.0| CA GEOM 13.0
22. ANS:
STA:
23. ANS:
STA:
24. ANS:
STA:
25. ANS:
STA:
26. ANS:
STA:
27. ANS:
STA:
28. ANS:
STA:
29. ANS:
STA:
30. ANS:
STA:
31. ANS:
STA:
32. ANS:
STA:
33. ANS:
STA:
34. ANS:
35. ANS:
36. ANS:
37. ANS:
38. ANS:
39. ANS:
40. ANS:
41. ANS:
42. ANS:
43. ANS:
44. ANS:
45. ANS:
46. ANS:
47. ANS:
48. ANS:
49. ANS:
C
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
C
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
C
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
B
REF: 3-5 The Polygon Angle-Sum Theorems
CA GEOM 12.0| CA GEOM 13.0
D
REF: 3-6 Lines in the Coordinate Plane
C
REF: 3-6 Lines in the Coordinate Plane
D
REF: 3-6 Lines in the Coordinate Plane
D
REF: 3-6 Lines in the Coordinate Plane
C
REF: 3-6 Lines in the Coordinate Plane
D
REF: 3-6 Lines in the Coordinate Plane
A
REF: 3-6 Lines in the Coordinate Plane
D
REF: 3-6 Lines in the Coordinate Plane
A
REF: 3-7 Slopes of Parallel and Perpendicular Lines
A
REF: 3-7 Slopes of Parallel and Perpendicular Lines
D
REF: 3-7 Slopes of Parallel and Perpendicular Lines
C
REF: 3-7 Slopes of Parallel and Perpendicular Lines
D
REF: 3-7 Slopes of Parallel and Perpendicular Lines
C
REF: 3-7 Slopes of Parallel and Perpendicular Lines
B
REF: 3-7 Slopes of Parallel and Perpendicular Lines
C
REF: 3-7 Slopes of Parallel and Perpendicular Lines
SHORT ANSWER
50. ANS:
a. Corresponding angles.
b. Vertical angles.
c. Transitive Property.
REF: 3-1 Properties of Parallel Lines
51. ANS:
a. Vertical angles.
b. Transitive Property.
c. Alternate Interior Angles Converse.
STA: CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
REF: 3-2 Proving Lines Parallel
52. ANS:
STA: CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
53. ANS:
m (interior) = 120
m (exterior) = 60
STA: CA GEOM 12.0| CA GEOM 13.0
REF: 3-5 The Polygon Angle-Sum Theorems
54. ANS:
y = –2x + 1
STA: CA GEOM 12.0| CA GEOM 13.0
y
6
4
2
–6
–4
–2
2
4
6
x
–2
–4
–6
REF: 3-6 Lines in the Coordinate Plane
ESSAY
55. ANS:
[4]
[3]
[2]
[1]
By the definition of , r  s implies m2 = 90, and t  s implies m6 = 90. Line s
is a transversal. 2 and 6 are corresponding angles. By the Converse of the
Corresponding Angles Postulate, r || t.
correct idea, some details inaccurate
correct idea, not well organized
correct idea, one or more significant steps omitted
REF: 3-3 Parallel and Perpendicular Lines
56. ANS:
STA: CA GEOM 7.0
[4]
[3]
[2]
[1]
correct idea, some details inaccurate
correct idea, some statements missing
correct idea, several steps omitted
REF: 3-2 Proving Lines Parallel
STA: CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
57. ANS:
[4] w + 31 + 90 = 180, so w = 59º. Since vertical angles are congruent, y = 59º. Since
supplementary angles have measures with sum 180, x = v = 121º. z + 68 + y = z
+ 68 + 59 = 180, so z = 53º.
[3] small error leading to one incorrect answer
[2] three correct answers, work shown
[1] two correct answers, work shown
REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem
STA: CA GEOM 12.0| CA GEOM 13.0
OTHER
58. ANS:
l and m are both perpendicular to n. Explanation: Since l and m are parallel,
are supplementary by the
Same-Side Interior Angles Theorem. It is given that
, so 180 = m1 + m2 = m1 + m1 = 2m1,
and m1 = 90 = m2. Since 1 and 2 are right angles, l is perpendicular to n and m is perpendicular to n.
REF: 3-1 Properties of Parallel Lines
STA: CA GEOM 2.0| CA GEOM 4.0| CA GEOM 7.0
59. ANS:
a. concave hexagon
b. concave dodecagon
c. not a polygon; two sides intersect between endpoints
REF: 3-5 The Polygon Angle-Sum Theorems
STA: CA GEOM 12.0| CA GEOM 13.0
60. ANS:
A polygon is convex if the points of all the diagonals are inside or on the polygon.
REF: 3-5 The Polygon Angle-Sum Theorems
STA: CA GEOM 12.0| CA GEOM 13.0