Parallel And Perpendicular Lines Worksheet Page 11
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50Parallel Lines and Angle Pairs
Theorem
Examples
3.1
Alternate Interior Angles Theorem
If two parallel lines
4
5
are cut by a transversal, then each pair of alternate
3
6
interior angles is congruent.
1
2
3.2
Consecutive Interior Angles Theorem
If two parallel
m 4
m 6
180
3
4
lines are cut by a transversal, then each pair of
m 3
m 5
180
5
6
consecutive interior angles is supplementary.
7
8
3.3
Alternate Exterior Angles Theorem
If two parallel lines
1
8
are cut by a transversal, then each pair of alternate
2
7
exterior angles is congruent.
You will prove Theorems 3.2 and 3.3 in Exercises 40 and 39, respectively.
a
Theorem 3.1
Proof
Proof
b
1
Given:
a b
;
p
a
b
is a transversal of
and
.
2
5
6
Prove:
2
7,
3
6
3
4
Paragraph Proof:
a b
We are given that
with a
7
8
p
p
transversal
. By the Corresponding Angles Postulate,
2
4 and
8
6. Also,
4
7 and
3
8
because vertical angles are congruent. Therefore,
2
7 and
3
6 since
congruence of angles is transitive.
A special relationship occurs when the transversal is a perpendicular line.
Theorem 3.4
Theorem 3.4
Perpendicular Transversal Theorem
In a plane, if a line is
m
perpendicular to one of two parallel lines, then it is
perpendicular to the other.
n
t
Theorem 3.4
Proof
Proof
1
p
Given:
p q
t
p
,
2
Prove:
t
q
q
Proof:
Statements
Reasons
p q
t
p
1.
,
1. Given
1 is a right angle.
2. Definition of
lines
2.
3. m 1
90
3. Definition of right angle
1
2
4. Corresponding Angles Postulate
4.
5. m 1
m 2
5. Definition of congruent angles
6. m 2
90
6. Substitution Property
7.
2 is a right angle.
7. Definition of right angles
t
q
8. Definition of
lines
8.
134 Chapter 3 Parallel and Perpendicular Lines
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