Download Parallel and Perpendicular Lines

Parallel and Perpendicular Lines
Angles and Parallel Lines
Class Work
1. Name all segments parallel to
:
2. Name all segments skew to
:
3. Name all segments intersecting with
:
4. Are segments
and
coplanar? Explain your answer.
5. Are segments
and
coplanar? Explain your answer.
Is each statement true always, sometimes, or never?
6. Two intersecting lines are skew.
7. Two parallel lines are coplanar.
8. Two lines in the same plane are parallel.
9. Two lines that do not intersect are parallel.
10. Two skew lines are coplanar
Classify each pair of angles as alternate interior, same-side interior, corresponding angles, or
none of these.
11.
12.
13.
14.
15.
16.
Angles and Parallel Lines
Homework
17. Name all segments parallel to
:
18. Name all segments skew to :
19. Name all segments intersecting with :
20. Are segments
and
coplanar? Explain your answer.
21. Are segments
and
coplanar? Explain your answer.
Classify each pair of angles as alternate interior, same-side interior, corresponding angles, or
none of these.
22. 7 and 12
23. 3 and 6
24. 6 and 11
25. 7 and 11
26. 4 and 10
27. 14 and 16
28. 2 and 3
29. 2 and 10
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State whether the following statements are always, sometimes, or never true:
30. Two coplanar lines are skew.
31. Two intersecting lines are in the same plane.
32. Two lines in the same plane are parallel.
Properties of Parallel Lines
Class Work
Use the given diagram to answer problems #33-41.
If m 9=540, then find the measure the following angles:
33.
34.
35.
36.
37.
If m
38.
39.
40.
41.
2=12x-54 and m
10=7x+26, then find the measure the following angles:
Find the values of the unknown variables in each figure. (# 42-46)
42.
43.
44.
45.
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46.
~2~
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Find measure of the following angles:
47.
48.
49.
50.
51.
State which segments (if any) are parallel.
52.
53.
54.
Properties of Parallel Lines
Homework
If m 9=620, then find the measure the following angles:
55.
56.
57.
58.
59.
If m
60.
61.
62.
63.
2=14x-24 and m
10=6x+72, then find the measure the following angles:
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Find the values of the unknown variables in each figure. (#64-68)
64.
65.
67.
66.
68.
Find measure of the following angles:
69.
70.
71.
72.
73.
State which segments (if any) are parallel.
74.
D
C
124°
124°
A
B
75.
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~4~
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76.
Equations of Lines and Slope
Class Work
Identify the slope of the line containing the given points:
77. (-2,11), (-5, 2)
78. (-4,11), (-4, -7)
79. (-2,22), (-5, 22)
Write the equation of the line in slope-y intercept form containing the given information:
80. m= 4; b=4
81. m= -2; (0,1)
82. (3, -12), (-5, 4)
83. (-16,-2), (-4, 7)
Write the equation of the line in standard form containing the following information:
84. (-7,4), (-7, 32)
85. (1,3), (6,-2)
86. x-intercept=8; y-intercept= -4
Write the equation of the line in point-slope form containing the following information:
87. m= -3, (-8,-2)
88. (5,2), (-3,4)
89. (-11,12), (-10, 11)
90. The cost (C) of a cab ride is $4 to start and $0.25 per tenth of a mile (t)
a. Write the equation that models the information.
b. How much will a 3 mile ride cost?
c. If you have $10, what is the greatest distance you can go?
91. The median house price (p) in Smallville in 2010 was $250,000. That value has grown
constantly at $5000 per year.
a. Write an equation that models the information.
b. How much will the median house be worth in 2020?
c. If this growth rate continues when will the median value reach $350,000?
92. The price(p) for the Sunnyside Day Camp to go to the movies is $7 for each adult and
$5 for each camper.
a. Write an equation that models the information.
b. How much will it cost for 5 adults and 20 campers?
c. If for every 6 campers, there needs to be an adult, how much will it cost for
40 campers?
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Equations of Lines and Slope
Homework
Identify the slope of the line containing the given points:
93. (-6,12), (-2, 5)
94. (14,11), (-14, 11)
95. (-2,17), (-2, 18)
Write the equation of the line in slope-y intercept form containing the given information:
96. m= -3; b=4
97. m= 6, (0,2)
98. (-2, -11), (3, 4)
99. (16,-2), (4, 6)
Write the equation of the line in standard form containing the following information:
100. (-7,41), (-9, 41)
101. (-5,12), (-5, 11)
102. x-intercept= -2; y-intercept= 6
Write the equation of the line in point-slope form containing the following information:
103. m= 5, (7,-1)
104. (6,2), (8,-4)
105. (-3,0), (-6,2)
106. The cost (C) of a cab ride is $5 to start and $0.30 per tenth of a mile (t)
a. Write the equation that models the information.
b. How much will a 4 mile ride cost?
c. If you have $20, what is the greatest distance you can go?
107. The median house price (p) in Largeton in 2010 was $300,000. That value has
grown constantly at $6000 per year.
a. Write an equation that models the information.
b. How much will the median house be worth in 2015?
c. If this growth rate continues when will the median value reach $360,000?
108. The price for the Sunnyside Day Camp to go to the movies is $8 for each adult
and $6 for each camper.
a. Write an equation that models the information.
b. How much will it cost for 4 adults and 15 campers?
c. If for every 5 campers, there needs to be an adult, how much will it cost for
36 campers?
Slopes of Parallel Lines
Class Work
109. Find an equation of the line in point-slope form passing through point (-2,5) and parallel to
the line whose equation is 4x -2y = -5
110. Two lines are represented by equations: 2x +4y =21 and y=kx -12.
What value of k will make lines parallel?
111. Find an equation of the line in slope-intercept form passing through point (-4,6) and parallel
to the line whose equation is y = -¾ x+11
112. The sides of a quadrilateral lie on the lines y= 4x +5, y= 1/3x +7, 8x - 2y= 1,
and x – 3y =2. Is the quadrilateral a parallelogram? Justify your answer.
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113. Is the following system of equations parallel? Justify your answer.
A
B
Slopes of Parallel Lines
Homework
114. Find an equation of the line in point-slope form passing through point (-3,2) and parallel to
the line whose equation is 6x -2y = 7.
115. Two lines are represented by equations: -4x +12y =21 and y=kx -12.
What value of k will make lines parallel?
116. Find an equation of the line in point-slope form passing through point (-8,3) and parallel to
the line whose equation is y = -¾ x+11.
117. The sides of a quadrilateral lie on the lines 3x+ y= 7, x + y= 12, 6x – 2y =2,
and x - y=2. Is the quadrilateral a parallelogram? Justify your answer.
118. Is the following system of equations parallel? Justify your answer.
A
B
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Perpendicular Lines and Angles
Class Work
Solve for the unknowns
119.
120.
In exercises 121-122 make a conclusion, how lines a and d are
related based on given information.( Lines a, b, c, and d are distinct and in the same plane. )
121.
122.
Perpendicular Lines and Angles
Homework
123.
124.
Lines a, b, c, and d are distinct lines in the same plane.
In exercises 125-126 make a conclusion, how lines b and d are
related based on given information.
125.
126.
Slopes of Perpendicular Lines
Class Work
127. Find an equation of the line passing through point (4,-5) and perpendicular to the
line whose equation is 3x -6y = -11.
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128. Two lines are represented by equations: -3x +6y =21 and y=kx +5.
What value of k will make lines perpendicular?
129. Find an equation of the line passing through point (8,-2) and perpendicular to the
line whose equation is y = 4x+11.
130. The sides of a quadrilateral lie on the lines y= 4x +5, y= 1/3x +7, x + 4y= 1,
and x – 3y =2, is the quadrilateral a rectangle? Justify your answer.
131. Is the following system of equations perpendicular? Justify your answer.
A
B
Slopes of Perpendicular Lines
Homework
132. Find an equation of the line passing through point (-6,2) and perpendicular to the line
whose equation is 4x +6y = -1
133. Two lines are represented by equations: 10x -15y =21 and y=kx +5.
What value of k will make lines perpendicular?
134. Find an equation of the line passing through point (8,-2) and perpendicular to the line
whose equation is y = -2x+11
135. The sides of a quadrilateral lie on the lines 4x - y= 5, x + 4y=7, 8x – 2y= 1, and
3x + 12y =2. Is the quadrilateral a rectangle? Justify your answer.
136. Is the following system of equations perpendicular? Justify your answer.
A
B
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Proofs Involving Parallel and Perpendicular Lines
Class Work
137. Fill in the blanks for the following two-column proof.
Given:
Prove : 𝑘䚫𝑚
Statements
1. p||n; <9䊃<5
2. <9䊃<1
3. <5䊃<1
4. k || m
Reasons
1.
2.
3.
4.
138. Fill in the blanks for the following paragraph proof.
0
Given:
Prove : 𝑝䚫𝑛
Given k || m, therefore m<12 + m<13 = 180°due to __1___.
It is also given that m<12+m<5 = 180°, therefore by
___2__ , m<12 + m<13 = m<12 + m<5. If we subtract
the m<12 from both sides by ___3__ we arrive at m<13 = m<5.
If m<13=m<5 then <13 @ <5 by __4___.
If the <13 @ <5 then by __5___ p || n.
Proofs Involving Parallel and
Perpendicular Lines
Homework
X
H
1
4
139. Write a 2 column proof.
Given: The diagram to the right
Prove: Vertical Angles Theorem
2
3
G
140. Write
suur asuuflow
r chart proof.
Given: AB || CD
Prove: m<4+m<2+m<5 = 180°
Y
B
A
1
4
C
Geometry Parallel & Perpendicular
~10~
2
3
5
D
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Constructing Parallel Lines
Class Work
141. Construct a line m that is parallel to line l that passes thru point C using the stated method.
Corresponding Angles
142. Error Analysis: A person was constructing the line n thru point D such that it
was parallel to line l using the alternate interior angles method. Using their markings,
state their mistake.
143. Use paper- folding techniques to construct parallel lines.
Constructing Parallel Lines
Homework
144. Error Analysis: A person was constructing the line n thru point D such that it
was parallel to line l using the alternate exterior angles method. Using their markings,
state their mistake.
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145. Construct parallel lines using a straightedge and compass using alternate interior angles.
146. Construct a 60° using a straightedge and compass.
147. Construct a regular hexagon inscribed in a circle using a straightedge and compass.
Constructing Perpendiculars
Class Work
148. Using a compass and straightedge, construct a perpendicular line to m through C.
149. A builder wants to put in the shortest access road possible to his new project.
If P represents his project and Elm Street is the road he is connecting to, where
should he build the road?
150. What did the person do incorrectly when constructing the perpendicular to m thru C?
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Constructing Perpendiculars
Homework
Construct a perpendicular line to m that passes thru point C.
151. Use a compass and straightedge.
152. Use paper folding to make the construction.
153. Rafee launches a toy rocket (R) straight up during his science lab. Since there
was no wind at that time, the rocket came back down to the same exact spot.
Construct the path of the rocket using a compass and straightedge.
154. What did the person do incorrectly when constructing the perpendicular to m thru C?
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Multiple Choice
1. Name the segment parallel to
a.
b.
c.
d.
2. Given the diagram
a. 32
b. 40
c. 86
d. 110
and skew to
and
, what is
3. Write the equation of the line in slope-y intercept form with m= 3 and b= -2.
a. y= -3x +2
b. y= -2x + 3
c. y= 3x + 2
d. y= 3x -2
4. Write the equation of the line in standard form with x-intercept=6 and y-intercept= -4.
a. 6x – 4y = 1
b. 2x – 3y = 12
c. x – y = 4
d. -4x + 6y = 1
5. Write the equation of the line in point-slope form that passes thru (5,2) and (-3,4).
a. y+4= ¼ (x-3)
b. y- 4= ¼ (x+3)
c. y-2= - ¼ (x+5)
d. y-2= - ¼ (x-5)
6. What is the equation of the line parallel to 2x+8y=10 and passes thru (-1,5)?
a. y – 5 = - ¼( x-1)
b. y – 5 = - ¼( x +1)
c. y – 5 = 4(x – 1)
d. y – 5 =4(x + 1)
7. What is the equation of the line perpendicular to y – 3 = 2/3(x + 3) and has an
x-intercept of 6?
a. y= 2/3x +4
b. y= 2/3x – 4
c. y= -3/2x + 6
d. y= -3/2x + 9
Geometry Parallel & Perpendicular
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In 8-9, use the diagram at the right.
8. Given
, what justifies 𝑘䚫𝑚?
a. Converse Alternate Interior Angles Theorem
b. Converse Alternate Exterior Angles Theorem
c. Converse Corresponding Angles Theorem
d. there is not enough info to state parallel
9. Given 𝑛䚫𝑝, what justifies
a. Alternate Interior Angles Theorem
b. Alternate Exterior Angles Theorem
c. Corresponding Angles Theorem
d. there is not enough info to make this statement
10. The cost © of school supplies is $3 per notebook (n) and $4 per packet of pens(p).
Which equation represents the total cost for pens and notebooks?
a. C= 4n + 3p
b. C= 7np
c. C= 4n + 3
d. C= 4p + 3n
11. The Fashion Club sells boosters for their spring show. Half page adds cost $12 and
full page adds $20. The printer donates the program so any ads sold are profit. If the
program made the club $188 and there were 7 full page ads, how many ½ page ads were
sold?
a. 4
b. 5
c. 6
d. 7
12. The distance from a point to a line can be found
a. along the arc of a circle
b. along the perpendicular to the line thru the point
c. calculating the opposite reciprocal of the y-intercept
d. constructing a circle centered at the point.
13. How many lines can be drawn through a point not on a line perpendicular to the given line?
a. none
b. one
c. two
d. infinitely many
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~15~
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Extended Response
1. Create a proof for:
Given:
Prove:
M
N
1
3
P
2
Q
2. Draw the line m on the graph provided so
that m passes thru (1,4) and (5,-3)
a. What is the equation of the line in point slope form?
b. Construct a parallel line n that contains (3,7)
c. What is the slope of line n?
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3. Draw the line m on the graph provided so that m passes thru (1,4) and (5,-3)
d. What is the equation of the line in standard form?
e. Construct a line p that is perpendicular that contains A(3,7)?
f. What is the slope of line p?
g. State how you would find the distance from A to m?
4. Using a compass and straightedge, construct parallel lines.
5. Using a compass and straightedge, construct a 60° angle.
Geometry Parallel & Perpendicular
~17~
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Answers
35. 126°
36. 54°
37. 54°
38. 138°
39. 42°
40. 42°
41. 138°
42. x= 144°
43. x= 64° and y= 49/4
44. x=6; z=2
45. x=24, y=11; z=22/5
46. x=33; y=2
47. 44°
48. 107°
49. 29°
50. 29°
51. 136°
52. Segments
and
are parallel
53. Segments
and
are parallel
54. None of these
55. 62°
56. 118°
57. 118°
58. 62°
59. 62°
60. 144°
61. 36°
62. 36°
63. 144°
64. x=55°
65. x=86° and y=7
66. x=9; y=6; z=7
67. x=15; y=10; z=8
68. x=25; y=3
69. 41°
70. 106°
71. 33°
1.
2.
3.
4.
Segments ,
,
,
Segments
Segments
Yes, because these segments are
parallel
5. No, these lines are skew, so they
are not coplanar.
6. Never
7. Always
8. Sometimes
9. Sometimes
10. Never
11. Same side interior
12. None of these
13. Alternate interior
14. Corresponding
15. Same-side interior
16. None of these
17. Segments
and
18. Segments
and
19. Segments
20. Yes, because they are parallel
21. No, these lines are skew, so they
are not coplanar
22. Corresponding
23. Same-side
24. Alternate interior
25. Corresponding
26. Corresponding
27. Same-side interior
28. None of these
29. None of these
30. Never
31. Always
32. Sometimes
33. 54°
34. 126°
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72. 33°
73. 129°
74. cannot be determined
75. Segments
and
are parallel
76. Segments
and
are parallel
77. 3
78. No slope (undefined)
79. 0
80. y= 4x +4
81. y= -2x+1
82. y= -2x-6
83. y= ¾x +10
84. x= -7
85. x +y =4
86. x – 2y = 8
87. y+2= -3(x+8)
88. y-2= -1/4(x-5) or y-4=-1/4(x+3)
89. y-12= -(x+11) or y-11=-(x+10)
90. a. C=.25t+4 b. $11.50 c. 2.4 mi
91. a. p-250000=5000(x-2010) b.
$300,000 c. 2030
92. a. 7A+5C=P b. $135 c. $249
93. -7/4
94. 0
95. No slope (Undefined)
96. y= -3x+4
97. y=6x +2
98. y=3x-5
99. y=-2/3 x +26/3
100.
y=41
101.
x=-5
102.
3x – y= -6
103.
y+1=5(x-7)
104.
y-2= -3(x-6) or y+4=-3(x-8)
105.
y= -2/3(x+3) or y-2=2/3(x+6)
106.
a. C=.30t+5 b. $17 c. 5 miles
107.
a. y-300,000=6000(x-2010)
b. $330,000 c. 2020
Geometry Parallel & Perpendicular
~19~
108.
a. 8A + 6C = P b. $122 c.
$280
109.
y -5= 2(x+2)
110.
k= -1/2
111.
y= -3/4x+3
112.
Yes,𝑦 = 4𝑥 + 5䚫8𝑥 −
1
2𝑦 = 1 ; 𝑦 = 3 𝑥 + 7䚫𝑥 − 3𝑦 =
2
113.
a. no; slope of m=2, slope of
n = 4/3; b. yes; slope of n and m =
-4/5
114.
y -2 = 3(x+3)
115.
k=1/3
116.
y-3=-3/4(x+8)
117.
no;
118.
a. no; slope of m = 1, slope of
n = 4/7; b. no, slope of m=-1/6,
slope of n= -1/7
119.
x=9 and y=8 and z=7
120.
x=8 and y=7
121.
a is perpendicular to d
122.
a is parallel to d
123.
x=6; y=12; z=7
124.
x=18; y=7
125.
b is perpendicular to d
126.
b is perpendicular to d
127.
y=-2x+3
128.
k= -2
129.
y=-1/4x
130.
no, y=4x +5 and y =1/3x +7
are not perpendicular
131.
a. no slope of m= -1/6; slope
of n= 7/2; b. no slope of m= 1/5;
slope of n= -1/10
132.
y=3/2x+11
133.
k=-3/2
134.
y=1/2x-6
135.
yes, slopes are m=-1/4 and 4
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136.
a. no slope of m= 3/2; slope
of n= -1/2; b. no, slope of m=1/5;
slope of n= -10
137.
Statement
P is parallel to n
Angle 9 is
congruent to
angle 5
Angle 9 is
congruent to
angle 1
Angle 5 is
congruent to
angle 1
K is parallel to
m
Reasoning
Given
Statement
<1 and <4 form a
linear pair.
<3 and <4 form a
linear pair
m<1+m<4=180
m<3+m<4=180
m<1+m<4=
m<3+m<4
m<1=m<3
<1 @ <3
Corresponding
angles postulate
Substitution
property
140.
suur suur
AB || CD
<1 @ 4
<3 @ <5
m<1=m<4
m<3=m<5
138.
1. Same-side interior angles
theorem
2. Substitution prop of equality
3. Subtraction prop of equality
4. Definition of congruent
segments
5. Converse of corresponding
angles postulate
Reasoning
Given
Linear Pair
Postulate
Substitution
property of =
Subtraction
prop of =
Def. of
congruent
Segments
Written as a flowchart.
Statement
converse of
corresponding
angles postulate
Geometry Parallel & Perpendicular
139.
m<1+m<2+m<3
=180
m<4+m<2+m<5
=180
Reasoning
Given
Alternate
interior angles
th.
Def. of
congruent
segments
Angle Addition
Postulate
Substition Prop
Of =.
141.
See student work
142.
made same side interior the
same
143.
See student work
144.
Made angles congruent that
should be supplementary.
145.
see student work
146.
see student work
147.
see student work
148.
see student work
~20~
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b. construct 䊃
149.
construct through P to
Elm.
150.
arcs weren’t drawn to
intersect
151.
see student work
152.
see student work
153.
see student work
154.
arcs weren’t drawn to
intersect
c. -7/4
3. a. 7x + 4y = 23
b. construct
c.
4/7
d. Find the equation of line p. Solve
the system of equations for the point of
intersection. Use the distance formula to
find the distance from A to the point of
intersection.
REVIEW
1. c
2. c
3. d
4. b
5. d
6. b
7. d
8. c
9. d
10. d
11. a
12. b
13. b
4. See student work
5. See student work
EXTENDED RESPONSE
1.
Statement
Reasoning
Given
Given
Alternate
Interior Angles
Theorem
Substitution
2.
a. y – 4 = -7/4(x – 1)
or y + 3 = -7/4(x – 5)
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~21~
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