Parallel and Perpendicular Lines

1
Chapter 3

• Parallel and Perpendicular Lines

2
Sec. 3-1

• Properties of Parallel Lines

Objective a) Identify Angles formed by
Two Lines a Transversal. b) To
Prove Use Properties of
Parallel Lines.
3
Parallel Lines Two lines in the same plane
which never intersect. Symbol //

• Transversal A line that intersects two //
  lines.
• 8 Special Angles are formed.

t
1
2
Interior Portion of the // Lines
m
3
4
5
6
n
8
7
4

• Corresponding Angles
• Most Important Angle Relationship
• Always Congruent
• Cut the Transversal lay the top part onto the
  bottom part. Overlapping Angles are
  Corresponding.

Corresponding Angles ?1 ?5 ?2 ?6 ?3
?7 ?4 ?8
1
2
3
4
5
6
7
8
5
P(3 1) Corresponding Angle Postulate
If a Transversal Intersects two // lines, then
the corresponding angles are Congruent.
1
2
3
4
6
5
8
7
6
1
2

• 4 Pairs of Vertical Angles
• Are Congruent
• ?1 ? ?4 ?2 ? ?3
• ?5 ? ?8 ?6 ? ?7

3
4
6
5
8
7
Special Interior Angles
?3 ?6 ?4 ?5
Alternate Interior Angles

• Are congruent

?3 ?5 ?4 ?6
Same-Sided Interior Angles

• Are Supplementary

( 180)
7
Th (3-1) Alternate Interior Angle Theorem
If a Transversal intersects two // lines, then
the alternate interior angles are congruent.
1
2
l
3
4
Given l // m Prove ?3 ? ?6
5
6
m
7
8

• Statements
• l // m
• ?3 ? ?7
• ?7 ? ?6
• ?3 ? ?6

• Reasons
• Given
• Corrsp. Angles are Congruent
• Def. of Vertical Angles
• Subs

8
Th (3-2) Same-Sided Interior Angle Theorem
If a Transversal intersects two // lines, then
the same-sided interior angles are supplementary.
1
2
l
3
4
Given l // m Prove ?4 ?6 are Supplementary
5
6
m
7
8

• Reasons
• Given
• ? Add. Postulate
• Corrsp. ?s are ?
• Subs
• Def of Supplementary

• Statements
• l // m
• m?4 m?2 180
• m?2 m?6
• m?4 m?6 180
• ?4 ?6 are Supplementary

9
Examples 1 2

• Solve for the missing ?s

• Solve for x, then for each ?.

14x - 5
l
l
5x - 20
3x
13x
m
m
14x 5 13x -5 -x 5 x
5x 20 3x 180 8x 200 x 25
10
Use what you have learned!
2. Solve for angles a, b, c if l//m

• 1. Find m?2 if l//m.

m
c
a
b
42?
l
1
2
65
40
m
l
m?a 65 (Alt. Inter. ?) m?c 40 (Alt. Inter.
?) m?a m?b m?c 180 65 m?b 40 180
m? 75
m?1 42 (Corrsp. ?) m?1 m?2 180 42 m?2
180 m?2 138
11
Solve for x and find the measure of each angle if
l//m.
l
(x 40)
x
m
x x 40 180 2x 40 180 -40
-40 2x 140 x 70