3.4 Proving Lines are Parallel

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3.4 Proving Lines are Parallel
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Postulate 16 Corresponding Angles Converse (pg
143 for normal postulate 15)
Goal 1 Proving Lines are Parallel

• If two lines are cut by a transversal so that
  corresponding angles are congruent, then the
  lines are parallel.

j
k
j ll k
3
Theorem 3.8 Alternate Interior Angles Converse
(pg 143 Theorem 3.4)

• If two lines are cut by a transversal so that
  alternate interior angles are congruent, then the
  lines are parallel.

j
3
1
k
4
Theorem 3.9 Consecutive Interior Angles
Converse (pg 143 Theorem 3.5)

• If two lines are cut by a transversal so that
  consecutive interior angles are supplementary,
  then the lines are parallel.

j
2
1
k
5
Theorem 3.10 Alternate Exterior Angles Converse
(pg 143 Theorem 3.6)

• If two lines are cut by a transversal so that
  alternate exterior angles are congruent, then the
  lines are parallel.

4
j
k
5
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Prove the Alternate Interior Angles Converse

• Given ?1 ? ?2
• Prove m n

3
m
2
1
n
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Example 1 Proof of Alternate Interior Converse

• Statements
• ?1 ? ?2
• ?2 ? ?3
• ?1 ? ?3
• m n

• Reasons
• Given
• Vertical Angles
• Transitive prop.
• Corresponding angles converse

Given ?1 ? ?2 Prove m n
3
m
2
1
n
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Example 2 Proof of the Consecutive Interior
Angles Converse

• Given ?4 and ?5 are supplementary
• Prove g h

g
6
5
4
h
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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.

g
6
5
Given ?4 and ?5 are supplementary Prove g h
4
h
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Find the value of x that makes j k.
Example 3 Applying the Consecutive Interior
Angles Converse

• 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

4x?
x?
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Goal 2 Using Parallel ConversesExample 4
Using CorrespondingAngles 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

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Solution

• Because corresponding angles are congruent,
  the boats paths are parallel. Parallel lines do
  not intersect, so the boats paths will not cross.

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Example 5 Identifying parallel lines

• Decide which rays are parallel.

H
E
G
61?
58?
62?
59?
C
A
B
D
A. Is EB parallel to HD? B. Is EA parallel to
HC?
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Example 5 Identifying parallel lines (cont.)

• Decide which rays are parallel.

H
E
G
61?
58?
B
D

• 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.

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Example 5 Identifying parallel lines (cont.)

• Decide which rays are parallel.

H
E
G
120?
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
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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.