chapter 3 parallel and perpendicular lines parallel and
TRANSCRIPT
ChapterChapterParallel andPerpendicular LinesParallel andPerpendicular Lines
104
33
Will the boats’ pathsWill the boats’ pathsWill the boats’ paths
ever cross?ever cross?ever cross?
If a sailboat heads directly into the wind,the sails flap and are useless. Onesolution is for boats to sail at an angle to the wind, as shown in the diagram.
During a race, sailboats often head intothe wind at the same angle and theyappear to be racing in parallel paths—paths that never cross.
Learn More About ItLearn More About ItLearn More About ItYou will learn more about paths ofsailboats in Exercise 25 on p. 149.
wind
45�
45�
wind
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Who usesParallel and Perpendicular Lines?
How will you use these ideas?
• Understand how escalators work. (p. 112)• Learn how a compass and map are used in orienteering.
(p. 119)• See how surfing and kite flying are combined in the sport of
kiteboarding. (p. 141)• Understand how the strings of a guitar are related to the
frets. (p. 148)
BICYCLE DESIGNERThe angles in a bike frameinfluence the position of therider. Georgena Terry buildsbicycles whose designsfeature a frame geometrymore suitable for women. (p. 124)
PHYSICAL THERAPISTPhysical therapists use angles to measure the range of motion of elbows,knees, and other joints. These measurements helpthe therapist evaluate people’s injuries. (p. 134)
105
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l
Sm
P
is not in plane S.
106 Chapter 3 Parallel and Perpendicular Lines
Chapter Study GuideStudy Guide33
What’s the chapter about?
• Identifying relationships between lines• Using properties of parallel and perpendicular lines• Showing that two lines are parallel
Key Words
Take this quick quiz. If you are unsure of an answer, look at the
reference pages for help.
Vocabulary Check (refer to p. 75)
1. Name a pair of vertical angles.
�A a1 and a4 �B a2 and a4
�C a2 and a3 �D a3 and a4
Skill Check (refer to pp. 75, 673)
2. The measure of a7 is 68�, and a7 and a8 form a linear pair. What is the measure of a8?
�F 22� �G 68� �H 112� �J 80�
3. Which of the following is a solution of the equation 5x � 9 � 6x � 11?
�A �12
10� �B 2 �C 11 �D 20
PREVIEW
12 34
PREVIEW
Chapter Readiness QuizPREPAREPREPARE
Reading and Drawing DiagramsVISUAL STRATEGYVISUAL STRATEGY
Visualize It!
• parallel lines, p. 108• perpendicular lines, p. 108• skew lines, p. 108• parallel planes, p. 109• transversal, p. 121
• corresponding angles, p. 121• alternate interior angles, p. 121• alternate exterior angles, p. 121• same-side interior angles, p. 121• converse, p. 136
Use dashed lines to show parts of afigure that are hidden behind otherparts. The dashed line shows thatline n is not in plane S.
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Think About It
Using your sketch from the steps
above, label the corners as shown.
Then extend AD&* and CG&*.
1. Will AD^&( and CG^&( ever intersect in space? Lines that intersect on the page do not necessarily intersectin space.
2. Determine whether the following pairs of lines will intersect in space.
a. BC^&( and AE^&( b. GH^&( and DH^&(
c. CD^&( and DH^&( d. AB^&( and EH^&(
Activity 107
Activity Lines in SpaceLines in SpaceLines in Space3.13.1
Explore
�1 Use a straightedge to drawtwo identical rectangles.
�2 Connect the correspondingcorners of the rectangles.
�3 Erase parts of “hidden” linesto form dashed lines.
Question
How are lines related in space?
Shading your sketchwill help make it lookthree-dimensional.
Visualize It!
• pencil• straightedge• lined paper
Materials
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Determine whether the lines are skew.
a. f and h
b. f and g
Solution
a. Lines f and h are not skew lines because they intersect.
b. Lines f and g are skew lines.
EXAMPLE 222 Identify Skew Lines
Two lines are if they lie in the same plane and do not intersect. On the building, lines rand s are parallel lines. You can writethis as r � s. Triangles ( ) are used to indicate that the lines are parallel.
Two lines are if they intersect to form a right angle. Lines s and t are perpendicular lines. You can write this as s ∏ t.
perpendicular lines
parallel lines
108 Chapter 3 Parallel and Perpendicular Lines
Goal
Identify relationshipsbetween lines.
Key Words
• parallel lines
• perpendicular lines
• skew lines
• parallel planes
• line perpendicular to a plane
3.13.1 Relationships Between Lines
Determine whether the lines are parallel,perpendicular, or neither.
a. n and m b. p and q c. n and p
Solution
a. Lines n and m are parallel.
b. Lines p and q are neither parallel norperpendicular.
c. Lines n and p are perpendicular.
EXAMPLE 111 Identify Parallel and Perpendicular Lines
Two lines are if they do not lie in the same plane. Skew lines never intersect. In the diagram, lines c and b are skew lines.
skew lines
B
f
g
h
m
p
n
q
A
c
b
Lines that intersect on the page do notnecessarily intersect in space, such as lines f and g.
B
f
gh
Visualize It!
r s
tt
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3.1 Relationships Between Lines 109
Identify Relationships Between Lines
Identify Relationships in Space
a. Name a plane that appears parallel to plane B.
b. Name a line that appears perpendicular to plane B.
Solution
a. Plane C appears parallel to plane B.
b. Line l appears perpendicular to plane B.
EXAMPLE 333 Identify Relationships in Space
Use the diagram.
1. Name a pair of parallel lines.
2. Name a pair of perpendicular lines.
3. Name a pair of skew lines.
Two planes are if they do not intersect.A is a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it.
line perpendicular to a planeparallel planes
Think of each segment in the diagram as part of a line.
4. Name a line that is skew to VW^&*(.
5. Name a plane that appears parallel to plane VWX.
6. Name a line that is perpendicular to plane VWX.
p
q
t
A
x
y
z
T
P
X
R V
U
S W
m
n
lD
C
B
F G
Plane F is parallel to plane G. Line t is perpendicular to plane A.
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110 Chapter 3 Parallel and Perpendicular Lines
1. How are skew lines and parallel lines alike? How are they different?
Fill in the blank with � or ∏ to make the statement true.
2. Line k __?__ line m.
3. Line m __?__ line l.
4. Line l __?__ line j.
5. Line k __?__ line j.
Match the photo with the corresponding description of the chopsticks.
A. skew B. parallel C. intersecting
6. 7. 8.
Line Relationships Determine whether the lines are parallel,
perpendicular, or neither.
9. a and c 10. q and s 11. y and z
Skew Lines Determine whether the lines are skew. Explain.
12. u and w 13. m and n 14. j and k
Practice and Applications
Skill Check
Vocabulary Check
Guided Practice
Exercises3.13.1
Example 1: Exs. 9–11Example 2: Exs. 12–14Example 3: Exs. 15–19,
21–24
Homework Help
Extra Practice
See p. 679.
ba
c
u
v
w
m
nl
j k
qr
s
xy
z
j
k
ml
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3.1 Relationships Between Lines 111
Identifying Relationships In Exercises 15–19, think of each
segment in the diagram as part of a line. Fill in the blank with
parallel, perpendicular, or skew.
15. DE^&(, AB^&(, and GC^&( appear to be __?__.
16. DE^&( and BE^&( are __?__.
17. BE^&( and GC^&( are __?__ .
18. BE^&( is __?__ to plane DEF.
19. Plane GAD and plane CBE appear to be __?__.
20. Tightrope Walking Philippe Petit sometimes uses a long pole tohelp him balance on the tightrope. Are the rope and the pole atthe left intersecting, perpendicular, parallel, or skew?
Relationships in Space Think of each segment in the diagram
below as part of a line.There may be more than one correct answer.
21. Name a line that appears parallel to QR^&(.
22. Name a line perpendicular to QR^&(.
23. Name a line skew to QR^&(.
24. Name a plane that appears parallel to plane QRS.
Sketch a figure that fits the description.
25. Three lines that are parallel
26. A line that is perpendicular to two parallel lines
27. Two planes that intersect
28. A line that is perpendicular to two parallel planes
29. A line that is perpendicular to two skew lines (Hint: Start bysketching a figure like the one above in Exercises 21–24.)
Furniture Design In Exercises 30–33,
use the photo of the chair shown at
the right.
30. Name two pairs of parallel lines.
31. What kind of lines are h and m?
32. Name two lines that are skew.
33. How many lines shown on the chair
are perpendicular to j?
Visualize It!
PHILIPPE PETIT, the Frenchhighwire artist, walked morethan 2000 feet up an inclinedcable to the Eiffel Tower. The photo above is from aperformance in New York City.
Balance
G C
FH
A B
D E
U
S
V
P RW
T
X
h
nj
k
l
m
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A prism is a three-dimensional figure with two identical faces, called bases, that lie in parallel planes as shown below.
a. Sketch a prism with bases that have six sides. Label the prism.
b. Name two edges that appear parallel.
Solution
a. To draw the prism, follow these steps.
�1 �2
b. Edges AG&* and DK&** appear parallel.
EXAMPLE Sketch a Prism
Escalators In Exercises 34 and 35, use the following information.
When a step on an up-escalator reaches the top, it flips over and goes back down to the bottom. On each step, let plane A be the plane you stand on.
34. As each step moves around theescalator, is plane A always parallel to the ground level? Explain.
35. When a person is standing on plane A, is it parallel toground level? Explain.
112 Chapter 3 Parallel and Perpendicular Lines
Prisms Sketch a prism with bases that have the given number of
sides. Label the prism. Name two edges that appear parallel.
36. Three sides 37. Four sides 38. Five sides
ground level
drive wheel
plane A
base
base
AB
M L
H J
C
F E D
G K
Sketch two identical bases inparallel planes. In this case,the bases have six sides.
Connect the bases and makehidden edges dashed. Labelthe prism.
Prism
VISUAL STRATEGY
Use dashed lines to show parts of afigure that are hiddenbehind other parts.
Student Help
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3.1 Relationships Between Lines 113
Challenge Fill in the blank with always, sometimes, or never.
39. Two skew lines are __?__ parallel.
40. Two perpendicular lines __?__ intersect.
41. Two skew lines are __?__ coplanar.
42. Multiple Choice Two lines are __?__ lines if they do not lie in the same plane and they do not intersect.
�A perpendicular �B parallel
�C coplanar �D skew
43. Multiple Choice Use the diagram below to determine which of the following statements is false. Think of each segment in the diagram as part of a line.
�F QP^&( and MP^&( are not parallel.
�G MP^&( and NR^&(are skew.
�H JM^&( and KS^&( are perpendicular.
�J Plane KJM and plane QPM are not parallel.
If-Then Statements Identify the hypothesis and the conclusion
of the if-then statement. (Lesson 2.5)
44. If the band plays, then each member gets $50.
45. If ma5 � 120�, then a5 is obtuse.
46. If there is a sale, then the store will be crowded.
47. If we can get tickets, then we’ll go to the movies.
Properties of Congruence Use the property to complete the
statement. (Lesson 2.6)
48. Reflexive Property of Congruence: __?__ c aXYZ
49. Symmetric Property of Congruence: If a1 c a2, then __?__c __?__.
50. Transitive Property of Congruence: If AB&* c EF&* and EF&* c ST&*,then __?__ c __?__.
Reciprocals Find the reciprocal. (Skills Review, p. 656)
51. 26 52. �7 53. 10 54. �83
�
Integers Evaluate. (Skills Review, p. 663)
55. 18 � (�3) 56. �4 � 2 57. 17 � (�6) 58. 16 � (�5)
59. �5 � 31 60. 24 � 28 61. (�8)(�10) 62. �25 � 19
Algebra Skills
Mixed Review
Standardized Test
Practice
P P
K
S
J
RMN
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