Download GETE0303

Parallel and
Perpendicular Lines
3-3
3-3
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To relate parallel and
Complete each statement with always, sometimes, or never.
perpendicular lines
Lesson 1-4
1. Two lines in the same plane are 9 parallel. sometimes
. . . And Why
1
To relate parallel and
perpendicular lines
Examples
2. Perpendicular lines 9 meet at right angles. always
To show how to fit sides of a
picture frame together, as in
Example 1
Objectives
1
2
3. Two lines in intersecting planes are 9 perpendicular. sometimes
Real-World Connection
Using Theorem 3-11
4. Two lines in parallel planes are 9 perpendicular. never
Math Background
Theorems 3-9, 3-10 and 3-11 are
generally easy for students to
remember. However, the phrase
in a plane often needs
clarification. Draw two parallel
lines on the board. At one point
on one of the lines, hold a pencil
or pointer perpendicular to the
chalkboard. Students should
recognize that the pointer is not
perpendicular to both lines
because the three lines are not
coplanar. A similar demonstration
can be done for Theorem 3-10
with a set of perpendicular lines
and a pointer. Finally, using a
pointer and two parallel lines one
can demonstrate that parallel
lines do not need to lie in the
same plane.
Key Concepts
E
L
U
The two diagrams suggest
ways to draw parallel lines.
You can draw them
(a) parallel to a given line, or
(b) perpendicular to a given
line. Theorems 3-9 and 3-10
guarantee that the lines you
draw are indeed parallel.
R
Relating Parallel and Perpendicular Lines
R
1
Theorem 3-9
a
If two lines are parallel to the same line,
then they are parallel to each other.
b
c
a6b
Theorem 3-10
In a plane, if two lines are perpendicular to the
same line, then they are parallel to each other.
m6n
t
m
More Math Background: p. 124C
n
Theorem 3-10 includes the phrase in a plane. On the other hand, Theorem 3-9 is true
for any three such lines, whether they are coplanar (Exercise 3) or noncoplanar.
Proof
See p. 124E for a list of the
resources that support this lesson.
Proof of Theorem 3-10
Study what is given, what you are to prove,
and the diagram. Then write a paragraph proof.
r
Given: r ' t, s ' t
1
s
2
PowerPoint
t
Bell Ringer Practice
Check Skills You’ll Need
Prove: r 6 s
Proof: &1 and &2 are right angles by the definition of perpendicular, so they are
congruent. Since corresponding angles are congruent, r 6 s.
Lesson 3-3 Parallel and Perpendicular Lines
Special Needs
Lesson Planning and
Resources
Below Level
L1
Have students draw lines on both sides of a stationary
ruler. Then have students move the ruler so only one
side is aligned with a line, and draw a third line.
Discuss why these lines are all parallel.
learning style: tactile
141
For intervention, direct students to:
Solving Linear Equations
Lesson 1-4: Example 2
Extra Skills, Word Problems, Proof
Practice, Ch. 1
L2
Have students demonstrate Theorems 3-9 and 3-10
using spaghetti or other classroom objects. Have them
use the corner of an index card to measure 90° angles.
learning style: tactile
141
Guided Instruction
1
Math Tip
Real-World
EXAMPLE
Connection
Woodworking To make a frame for a painting,
a miter box and a backsaw are used to cut the framing
at 458 angles. Explain why
cutting the framing at this angle
ensures that opposite sides of
the frame will be parallel.
corners cut
to form 45⬚
Two adjacent 458 angles form
angles
a 908 angle. The opposite sides of
the frame are perpendicular to the
same side. Thus the opposite sides
are parallel because two lines
perpendicular to a third line are parallel.
Ask: When are all the angles
formed by a transversal
congruent? When the transversal
is perpendicular to parallel lines.
PowerPoint
Additional Examples
1 Suppose that the top and
bottom pieces of a picture frame
are cut to make 60° angles with
exterior sides of the frame. The
two sides should be cut at what
angle to ensure that opposite
sides of the frame will be
parallel? 30°
Quick Check
1 Can you assemble the framing at the right
into a frame with opposite sides parallel?
Explain. Yes; 30° ± 60° ≠ 90°
2 Study what is given, what you
are to prove, and the diagram.
Then write a paragraph proof.
60⬚
60⬚
60⬚
60⬚
30⬚
30⬚
Theorems 3-9 and 3-10 gave conditions by which you can conclude that lines are
parallel. Theorem 3-11 provides a way for you to conclude that lines are
perpendicular. You will prove Theorem 3-11 in Exercise 11.
s
a
Key Concepts
Theorem 3-11
n
In a plane, if a line is perpendicular to one of two
parallel lines, then it is also perpendicular to the other.
b
ᐉ
m
n'm
c
Proof
Given: In a plane, a s, c s,
and a 6 b
Prove: c 6 b
Proof: Lines a and v are both to
line s, so a n c because two lines
perpendicular to the same line
are parallel. It is given that a n b.
Therefore, c n b because two
lines parallel to the same line are
parallel to each other.
Using Theorem 3-11
a
c
b
Given: In a plane, a ' b, b ' c, and c ' d.
Prove: a ' d
d
Proof: Lines a and c are both perpendicular to line b, so a 6 c because two lines
perpendicular to the same line are parallel. It is given that c ' d . Therefore,
a ' d because a line that is perpendicular to one of two parallel lines is
perpendicular to the other (Theorem 3-11).
Quick Check
Closure
142
EXAMPLE
Study what is given, what you are to prove,
and the diagram. Then write a paragraph proof.
Resources
• Daily Notetaking Guide 3-3 L3
• Daily Notetaking Guide 3-3—
L1
Adapted Instruction
Name two methods this lesson
gives to prove that two lines are
parallel. Show that two lines are
parallel to the same line or
that in a plane two lines are
perpendicular to the same line.
2
142
2 From what is given in Example 2, can you also conclude b 6 d? Explain.
Yes; b n d by Thm. 3-10.
A
* ) * )
In the rectangular solid shown here, AC and BD
* )
* )
D
C
are parallel. EC is perpendicular
to AC , but it is
* )
not perpendicular to BD . This is why Theorem 3-11
G
states that the lines must be “in a plane.”
E
F
B
H
Chapter 3 Parallel and Perpendicular Lines
Advanced Learners
English Language Learners ELL
L4
After students have examined Theorem 3-9, ask: Is the
relationship “is parallel to” reflexive, symmetric,
and/or transitive? symmetric and transitive
learning style: verbal
For Example 1, show a miter box and a backsaw
borrowed from the woodworking teacher, or find
pictures from a tool catalog. Students need to
understand that these tools can make 45° cuts.
learning style: visual
EXERCISES
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
2. Practice
Practice and Problem Solving
Assignment Guide
A
Practice by Example
1. A carpenter is building a trellis for vines
to grow on. The completed trellis will have two
D
sets of overlapping diagonal slats of wood.
a. What must be true of &1, &2, and &3 if slats
A, B, and C must be parallel? l1 O l2 O l3
A
b. The carpenter attaches slat D so that it is
1
perpendicular to slat A. Is slat D perpendicular
to slats B and C? Justify your answer.
See back of book.
2. Study what is given, what you are to prove, and the
diagram. Then write a proof. See margin.
b
Example 1
(page 142)
GO for
Help
Example 2
Proof
(page 142)
1 A B 1-15
B
2
C
3
B
Apply Your Skills
c
For a guide to solving
Exercise 3, see p. 145.
Test Prep
Mixed Review
24-25
26-29
To check students’ understanding
of key skills and concepts, go over
Exercises 1, 2, 8, 12, 13.
d
a
3. Developing Proof Copy and complete this paragraph proof of Theorem 3-9 for
three coplanar lines.
GO for Help
14-23
Homework Quick Check
Given: In a plane, a ' b, b ' c, and c 6 d.
Prove: a 6 d
C Challenge
Exercises 3 Theorem 3-9 is also
true when the three lines are
noncoplanar, but that proof is
beyond the scope of the course.
If two lines are parallel to the same line, then they are parallel to each other.
Tactile Learners
Given: O 6 k and m 6 k
Exercise 13 Provide a physical
object such as a shoebox to help
students explain.
1
Prove: O 6 m
Proof: O 6 k means that &2 > &1 by
the a. 9 Postulate. m 6 k means that
b. 9 > c. 9 for the same reason.
By the Transitive Property of Congruence,
Converse of Corr. ' &2 > &3. By the d. 9 Postulate, O 6 m.
2
k
corr. '
l1, l3 (any order)
ᐉ
3
m
GPS Guided Problem Solving
L3
L4
Enrichment
Each of the following statements describes a ladder. What can you conclude about
the rungs, one side, or both sides of each ladder? Explain. 4-10. See margin.
4. The rungs are each perpendicular to one side.
L2
Reteaching
L1
Adapted Practice
Practice
Name
5. The rungs are parallel and the top rung is perpendicular to one side.
Class
Practice 3-3
L3
Date
Parallel Lines and the Triangle Angle-Sum Theorem
Find the value of each variable.
6. The sides are parallel. The rungs are perpendicular to one side.
1.
2.
3
yⴗ
65ⴗ
xⴗ
7. The rungs are perpendicular to one side. The other side is perpendicular to the
top rung.
30ⴗ
60ⴗ
nⴗ
39ⴗ
75ⴗ 68ⴗ
4.
5.
93ⴗ
6.
61ⴗ
46ⴗ
mⴗ
36ⴗ
79ⴗ
8. Each side is perpendicular to the top rung.
7.
8.
10ⴗ
55ⴗ
aⴗ bⴗ cⴗ
9. Each rung is parallel to the top rung.
44ⴗ
9.
28ⴗ
zⴗ
xⴗ
pⴗ
xⴗ
25ⴗ
53ⴗ
yⴗ
wⴗ
56ⴗ
62ⴗ
vⴗ tⴗ
Find the measure of each numbered angle.
10. The rungs are perpendicular to one side. The sides are not parallel.
10.
11.
1
12.
2
32ⴗ
Proof
11. Prove Theorem 3-11: In a plane, if a line is
GPS perpendicular to one of two parallel lines, then it is
also perpendicular to the other. See margin.
a
c
Given: In a plane, a ' b, and b 6 c.
Real-World
Connection
This ladder’s rungs are
perpendicular to each side.
Therefore, the rungs are
parallel to each other.
b
Prove: a ' c
Proof
2. a is perp. to b and c is
perp. to b, so a is
parallel to c because
two lines perp. to the
same line are parallel. d
is parallel to c and a is
parallel to c, so by Thm.
3-9, a is parallel to d.
13.
Lesson 3-3 Parallel and Perpendicular Lines
4. The rungs are parallel to
each other because they
are all perpendicular to
the same side.
5. All of the rungs are
perpendicular to one
side. The side is perp. to
the top rung, and
143
because all of the rungs
are parallel to each
other, the side is perp.
to all of the rungs.
6. The rungs are
perpendicular to both
sides. The rungs are
perp. to one of two
14.
2
69.7ⴗ
3
15.
3
4
5
38ⴗ
31ⴗ
46ⴗ
126.8ⴗ
70ⴗ
72ⴗ
86ⴗ
120ⴗ
116ⴗ 1
2
16. The sides of a triangle are 10 cm, 8 cm, and 10 cm. Classify the triangle.
17. The angles of a triangle are 44°, 110°, and 26°. Classify the triangle.
Use a protractor and a centimeter ruler to measure the angles and the sides
of each triangle. Classify each triangle by its angles and sides.
18.
12. Prove: If a line is perpendicular to each of two other lines, all in one plane, then
the two other lines are parallel.
s
r
2
Given: t ' r, t ' s
1
t
Prove: r 6 s
This is a restatement of Thm. 3-10.
lesson quiz, PHSchool.com, Web Code: aua-0303
© Pearson Education, Inc. All rights reserved.
140ⴗ
19.
20.
parallel lines, so they
are perp. to both lines.
7. The rungs are parallel to
each other because they
are all perpendicular to
one side. The sides are
parallel because they
are both perpendicular
to one rung.
8. The sides are parallel
because they are both
perpendicular to one
rung.
143
4. Assess & Reteach
PowerPoint
Lesson Quiz
B
A
13. Writing Theorem 3-10: In a plane, two lines
perpendicular to the same line are parallel.
Use the rectangular solid at the right to
explain why the words in a plane are needed.
See margin.
D
C
H
G
E
Complete the proof.
C
Challenge
a
b
x
y
GO
z
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0303
Given: In a plane, a y, b x,
a 6 y, and y 6 z
Prove: x 6 z
Proof: Because 1. a y and a 6 b,
line y is perpendicular to line b by
Theorem 2. 3-11. Lines x and y are
both perpendicular to line b, so
3. x n y because two lines
perpendicular to the same line
are 4. parallel. Therefore, x 6 z
because two lines that are
5. parallel to the same line are
parallel to each other.
a, b, c, and d are distinct lines in the same plane. Exercises 14-21 show different
combinations of relationships between a and b, b and c, and c and d. For each
combination of the three relationships, how are a and d related?
14. a 6 b, b 6 c, c 6 d a n d
15. a 6 b, b 6 c, c ' d a ' d
16. a 6 b, b ' c, c 6 d a ' d
17. a ' b, b 6 c, c 6 d a ' d
18. a 6 b, b ' c, c ' d a n d
19. a ' b, b 6 c, c ' d a n d
20. a ' b, b ' c, c 6 d a n d
21. a ' b, b ' c, c ' d a ' d
Critical Thinking The Reflexive, Symmetric, and Transitive Properties for
Congruence (O) are listed on page 105. 22–23. See margin.
22. Write reflexive, symmetric, and transitive statements for “is parallel to” ( 6 ).
State whether each statement is true or false and justify your answer.
23. Repeat Exercise 22 for “is perpendicular to” ( ').
Test Prep
Multiple Choice
24. In a plane, line e is parallel to line f, line f is parallel to line g, and line h is
perpendicular to line e. Which of the following CANNOT be true? C
A. e 6 g
B. h 6 f
C. g 6 h
D. e 6 h
* )
* )
Alternative Assessment
Have students draw and label
three parallel lines and a
perpendicular transversal.
Students identify one or two
angles as right and two lines as
parallel. They then write a proof
statement that requires use of the
theorems from the lesson.
Students trade papers with a
partner and write paragraph
proofs.
25. In a plane, AB is parallel to CD . &ABC is a right angle. What can you
conclude? G
* )
* )
I. AB ' BC
F. I only
* )
* )
II. BC ' CD
G. I and II
H. II only
GO for
Help
* )
Lesson 3-2 x 2 Algebra Determine the value of x for which a n b.
26.
53
27.
a
a
b
46
124⬚
b
Test Prep
Resources
Lesson 2-2
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 193
• Test-Taking Strategies, p. 188
• Test-Taking Strategies with
Transparencies
44⬚
(2x + 18)⬚
Each conditional statement below is true. Write its converse. If the converse is also
true, combine the statements into a biconditional.
28. If x 5 7, then x2 5 49.
144
144
* )
III. BC ' AD
J. I, II and III
Mixed Review
(3x ⫺ 2)⬚
9. All of the rungs are
parallel. All of the rungs
are parallel to one rung,
so they are all parallel to
each other.
F
If x2 ≠ 49, then x ≠ 7.
29. If two lines in a plane do not meet, then the lines are parallel.
If two lines in a plane are parallel, they do not meet. Two lines in a
plane do not meet if and only if the lines are parallel.
Chapter 3 Parallel and Perpendicular Lines
10. The rungs are parallel
because they are all
perpendicular to one side.
11. In the diagram, a b
means the marked l is a
rt. l. b n c means that
the corres. l formed by
a and c is a rt. l, so
a c.
13. Answers may vary.
Sample: In the diagram,
AB BH and AB BD,
but BH o BD. They
intersect.
22. Reflexive: a n a; false;
any line intersects itself.
Symmetric: If a n b, then
b n a; true; b and a are
coplaner and do not
intersect.
Transitive: In general, if
a n b, and b n c; then a n c;