Theorems About Lines and Angles
Rule

Converse Alternate Exterior Angles Theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.
Two parallel lines cut by a transversal forming two pairs of congruent angles
Based on the properties of the diagram, the following relation holds true.

If or then

Proof

The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate exterior angles and one more angle.

One pair of alternate exterior angles
It needs to be proven that and are parallel lines. It is already given that is congruent to
From the diagram, it can also be noted that and are vertical angles. By the Vertical Angles Theorem, these angles are congruent.
By the Transitive Property of Congruence, because is congruent to and is congruent to is congruent to
Further, and are corresponding angles. Hence, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem

If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since and are corresponding congruent angles, and are parallel lines. Each step of the proof will now be summarized in a two-column proof.

0.
Statement
0.
Reason
1.
1.
Given
2.
2.
Vertical Angles Theorem
3.
3.
Transitive Property of Congruence
4.
4.
Converse Corresponding Angles Theorem
Exercises