Theorems About Lines and Angles
Rule

Converse Alternate Interior Angles Theorem

If two lines and a transversal form alternate interior 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 characteristics 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 interior 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
The diagram shows that and are vertical angles. By the Vertical Angles Theorem, these angles are congruent.
Notice the common angle of in both relationships. By the Transitive Property of Congruence, since is congruent to and is congruent to then is congruent
The diagram also shows that and are corresponding angles. Given that relation, 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, then and are parallel lines. To summarize, all of the steps will be described 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