3.3: Proving Lines
Parallel
1
Today’s Objectives

Determine whether two lines are parallel
Converse

Converse – reverse of a statement

Example:
Original Statement:
If I have a fever, then I will not go to school.
Converse:
If I do not go to school, then I have a fever.
Compare

Original Postulate:

If a transversal intersects two parallel lines, then
corresponding angles are congruent.

Converse:

If corresponding angles are congruent, then the two lines
are parallel.
Converse of Corresponding
Angles Postulate

Postulate 3.2 – if two lines cut transversal have
congruent corresponding angles, then the two lines are
parallel
IF:
1
3
2
< 𝟏 ≅< 𝟓,
4
< 𝟐 ≅< 𝟔,
5
7
6
8
THEN:
< 𝟑 ≅< 𝟕,
𝒐𝒓…
< 𝟒 ≅< 𝟖.
THE LINES
ARE
PARALLEL
Converse of the Alternate
Interior Angles Theorem

Theorem 3.4 - If two lines cut by a transversal have
congruent alternate interior angles, then the lines are
parallel.
IF:
1
3
2
< 𝟑 ≅< 𝟔,
4
OR
5
7
6
8
THEN:
< 𝟒 ≅< 𝟓.
THE LINES
ARE
PARALLEL
Converse of the Consecutive
Interior Angles Theorem

Theorem 3.5 – If two lines cut by a transversal have
consecutive interior angles that are supplementary,
then the two lines are parallel.
IF:
1
3
5
7
2
4
𝒎 < 𝟓 + 𝒎 < 𝟑 = 𝟏𝟖𝟎,
OR
𝒎 < 𝟒 + 𝒎 < 𝟔 = 𝟏𝟖𝟎.
6
8
THEN:
THE LINES ARE PARALLEL
Converse of the Alternate
Exterior Angles Theorem

Theorem 3.6 – If two lines cut by a transversal have
congruent alternate interior angles, then the lines are
parallel.
IF:
1
3
2
< 𝟏 ≅< 𝟖,
4
𝒐𝒓…
5
7
6
8
THEN:
< 𝟐 ≅< 𝟕.
THE LINES
ARE
PARALLEL
Example

How do we know these lines are parallel?
Consecutive
Interior Converse
Example

How do we know these lines are parallel?
Alternate Exterior
Angles Converse
Example
Corresponding Angles Converse

What value of x makes these two lines parallel?
Corresponding
Congruent
3𝑥 = 60
𝑥 = 20
Example

Consecutive Interior
Angles Converse
What value of x makes these two lines parallel?
Same-Side Interior 
Supplementary
4𝑥 − 42 + 3𝑥 + 12 = 180
7𝑥 − 30 = 180
7𝑥 = 210
𝑥 = 30
Example

Alternate Interior
Angles Converse
Find the value of x that makes the two lines parallel.
Alternate Interior
Congruent
5𝑥 − 54 = 3𝑥 + 16
5𝑥 = 3𝑥 + 70
2𝑥 = 70
𝑥 = 35
Example

Consecutive Interior
Angles Converse
Are these two lines parallel?
 Therefore, the lines are parallel
Vertical Angles
135°
135°
45°
45°
135°
45°
Take Home Message

Converse – reverse of a statement

To prove lines are parallel  converse