3.4 Proving Lines are Parallel
Mrs. Spitz
Fall 2005
Standard/Objectives:
Standard 3: Students will learn and
apply geometric concepts
Objectives:
• Prove that two lines are parallel.
• Use properties of parallel lines to
solve real-life problems, such as
proving that prehistoric mounds
are parallel.
• Properties of parallel lines help you
predict.
HW ASSIGNMENT:
• 3.4--pp. 153-154 #1-28
Quiz after section 3.5
Postulate 16: Corresponding Angles
Converse
• If two lines are cut by a transversal
so that corresponding angles are
congruent, then the lines are
parallel.
Theorem 3.8: Alternate Interior
Angles Converse
• If two lines are cut by a transversal
so that alternate interior angles are
congruent, then the lines are
parallel.
Theorem 3.9: Consecutive
Interior Angles Converse
• If two lines are cut by a transversal
so that consecutive interior angles
are supplementary, then the lines
are parallel.
Theorem 3.10: Alternate
Exterior Angles Converse
• If two lines are cut by a transversal
so that alternate exterior angles
are congruent, then the lines are
parallel.
Prove the Alternate Interior
Angles Converse
Given: 1  2
Prove: m ║ n
3
2
1
m
n
Example 1: Proof of
Alternate Interior Converse
Statements:
1.
2.
3.
4.
1  2
2  3
1  3
m║n
Reasons:
1.
2.
3.
4.
Given
Vertical Angles
Transitive prop.
Corresponding
angles converse
Proof of the Consecutive
Interior Angles Converse
Given: 4 and 5 are supplementary
Prove: g ║ h
g
6 5
4
h
Paragraph Proof
You are given that 4 and 5 are
supplementary. By the Linear Pair
Postulate, 5 and 6 are also
supplementary because they form a
linear pair. By the Congruent
Supplements Theorem, it follows
that 4  6. Therefore, by the
Alternate Interior Angles Converse,
g and h are parallel.
Find the value of x that makes j ║ k.
Solution:
Lines j and k will be
parallel if the
marked angles are
supplementary.
x + 4x = 180 
5x = 180 
X = 36 
4x = 144 
So, if x = 36, then j
║ k.
x
4x
Using Parallel Converses:
Using Corresponding Angles
Converse
SAILING. If two boats sail at a 45
angle to the wind as shown, and
the wind is constant, will their
paths ever cross? Explain
Solution:
Because corresponding angles are
congruent, the boats’ paths are
parallel. Parallel lines do not
intersect, so the boats’ paths will
not cross.
Example 5: Identifying parallel
lines
Decide which rays are parallel.
H
E
62
A
58
G
61
59
B
C
A. Is EB parallel to HD?
B. Is EA parallel to HC?
D
Example 5: Identifying parallel
lines
Decide which rays are parallel.
H
E
58
B
G
61
D
A. Is EB parallel to HD?
mBEH = 58
m DHG = 61 The angles are
corresponding, but not congruent,
so EB and HD are not parallel.
Example 5: Identifying parallel
lines
Decide which rays are parallel.
H
E
120
A
G
120
C
A. B. Is EA parallel to HC?
m AEH = 62 + 58
m CHG = 59 + 61
AEH and CHG are congruent
corresponding angles, so EA ║HC.
Conclusion:
Two lines are cut by a transversal.
How can you prove the lines are
parallel?
Show that either a pair of alternate
interior angles, or a pair of
corresponding angles, or a pair of
alternate exterior angles is
congruent, or show that a pair of
consecutive interior angles is
supplementary.