Honors Geometry

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Honors Geometry

• Lesson 3.5
• Properties of Parallel Lines

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What You Should LearnWhy You Should Learn It

• Goal 1 How to identify angles formed by two
  lines and a transversal
• Goal 2 How to use properties of parallel lines
• You can apply the properties of parallel lines to
  many situations such as the parallel rays of
  sunlight entering a pane of glass.

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Angles Formed by a Transversal

• Transversal a line that intersect two or more
  coplanar lines at different points
• In the figure, the transversal t intersects the
  lines L and M

t
L
M
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Corresponding Angles

• Two angles are corresponding angles if they
  occupy corresponding positions, such as

t
4
1
3
2
L
M
5
8
6
7
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Alternate Interior Angles

• Two angles are alternate interior angles if they
  lie between L and M on opposite sides of t, such
  as

t
4
1
3
2
L
M
5
8
6
7
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Alternate Exterior Angles

• Two angles are alternate exterior angles if they
  lie outside L and M on opposite sides of t, such
  as

t
4
1
3
2
L
M
5
8
6
7
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Consecutive Interior Angles

• Two angles are consecutive interior angles if
  they lie between L and M on the same side of t,
  such as

t
4
1
3
2
L
M
5
8
6
7
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Example 1Naming Pairs of Angles

• How is related to the other angles?

n
4
1
m
3
2
10
9
11
12
8
7
5
L
6
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Example 1Naming Pairs of Angles

• How is related to the other angles?
• are a linear pair. So are
• are vertical angles
• are alternate exterior
  angles. So are
• are corresponding angles. So
  are

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Postulate 15Corresponding Angles Postulate

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

L
1
M
2
t
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Theorem 3.6Alternate Interior Angles Theorem

• If two parallel lines are cut by a transversal,
  then the pairs of alternate interior angles are
  congruent

L
1
3
M
2
4
t
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Theorem 3.7Consecutive Interior Angles Theorem

• If two parallel lines are cut by a transversal,
  then the pairs of consecutive interior angles are
  supplementary

L
1
3
M
2
4
t
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Theorem 3.8Alternate Exterior Angles Theorem

• If two parallel lines are cut by a transversal,
  then the pairs of alternate exterior angles are
  congruent

L
1
3
M
4
2
t
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Theorem 3.9Perpendicular Transversal Theorem

• If a transversal is perpendicular to one of two
  parallel lines, then it is perpendicular to the
  second.

L
M
t
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Proof of Theorem 3.6
3
1
2
L1
t
L2
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Proof of Theorem 3.6
3
1
2. Vertical Angles are congruent
2
L1
s
t
L2
4. Transitive Property of congruence
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The End