3.4 Proving Lines are Parallel

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3.4 Proving Lines are Parallel

• Mrs. Spitz
• Fall 2005

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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.

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HW ASSIGNMENT

• 3.4--pp. 153-154 1-28
• Quiz after section 3.5

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Postulate 16 Corresponding Angles Converse

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

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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.

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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.

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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.

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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

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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.

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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.

4x?
x?
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Using Parallel ConversesUsing 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

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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

• 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

• 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.