Download Geometry Section 3.2 Notes

CHAPTER
3
3.2 Transversals and
Angles
Copyright © 2014 Pearson Education, Inc.
Slide 1
Definitions
A transversal is a line that intersects two or more
coplanar lines at different points. The figure below
shows the eight angles formed by a transversal t and
two lines / and m.
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Slide 2
Definitions
Angles 3, 4, 5, and 6 lie between l and m. They are
interior angles.
Angles 1, 2, 7, and 8 lie outside of l and m. They are
exterior angles.
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Slide 3
Definitions
Alternate interior angles are nonadjacent interior
angles that lie on opposite sides of the transversal.
4 and 6
3 and 5
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Slide 4
Definitions
Same-side interior angles are interior angles that
lie on the same side of the transversal (sometimes
called consecutive interior angles).
4 and 5
3 and 6
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Slide 5
Definitions
Corresponding angles lie on the same side of the
transversal and in corresponding positions.
1 and 5
4 and 8
2 and 6
3 and 7
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Slide 6
Definitions
Alternate exterior angles are nonadjacent exterior
angles that lie on opposite sides of the transversal.
1 and 7
2 and 8
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Slide 7
Example
Identifying Angle Pairs
List all angle pairs in the figure.
a. alternate interior
b. corresponding
Solution
b. 7 and 3
a. 8 and 5
2 and 3
1 and 5
8 and 4
2 and 6
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Slide 8
Example
Classifying an Angle Pair
The photo shows the Hearst
Building in New York City.
The new tower (showing many
triangles) was completed in 2006.
Fill in the blank.
alternate exterior
a. 1 and 5 are _____________
angles.
same-side interior
b. 2 and 7 are _____________
angles.
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Slide 9
Theorem 3.2 Alternate Interior Angles
Theorem
Theorem If two lines are cut by a transversal and a
pair of alternate interior angles are congruent, then
the two lines are parallel.
If…
Then…
4  6
m
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Slide 10
Theorem 3.3 Corresponding Angles Theorem
Theorem If two lines are cut by a transversal and a
pair of corresponding angles are congruent, then the
lines are parallel.
If…
Then…
2  6
m
Copyright © 2014 Pearson Education, Inc.
Slide 11
Theorem 3.4 Alternate Exterior Angles
Theorem
Theorem If two lines are cut by a transversal and a
pair of alternate exterior angles that are congruent,
then the two lines are parallel.
If…
Then…
1  7
m
Copyright © 2014 Pearson Education, Inc.
Slide 12
Theorem 3.5 Same-Side Interior Angles
Theorem
Theorem If two lines are cut by a transversal are
two interior angles on the same side of the
transversal are supplementary, then the two lines are
parallel.
If…
Then…
m3  m6  180
m
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Slide 13
Theorem 3.6 Alternate Interior Angles
Converse (Converse of Theorem 3.2)
Theorem If two parallel lines are cut by a
transversal, then alternate interior angles are
congruent.
If…
Then…
m
4  6
3  5
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Slide 14
Example
Identifying Parallel Lines
Which lines are parallel if 1  7?
Justify your answer.
Solution
1 and 7 are not formed by line l, so we
concentrate on line a and line b with transversal m.
1 and 7 are alternate interior angles. If
1  7, then a || b by the Alternate Interior Angles
Theorem.
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Slide 15
Determining Whether Lines Are
Example
Parallel
The fence gate at the right is made up of pieces of
wood arranged in various directions. Suppose
1  2. Are lines r and s parallel? Explain.
Solution
Yes, r || s. 1 and 2 are
alternate exterior angles. If two
lines and a transversal form
congruent alternate exterior
angles, then the lines are
parallel (Alternate Exterior Angles Theorem).
Copyright © 2014 Pearson Education, Inc.
Slide 16
Using Algebra to Prove Lines Are
Example
Parallel
What is the value of x that makes a || b?
Solution
The two angles are same-side
interior angles. By the Same-Side
Interior Angles Theorem, a || b if
the angles are supplementary.
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Slide 17
Using Algebra to Prove Lines Are
Example
Parallel
What is the value of x that makes a || b?
(2x + 9) + 111 = 180
2x + 120 = 180
2x = 60
x = 30
Thus, if x = 30, then a || b.
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Slide 18