Converse Alternate Interior Angles Theorem

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.

It needs to be proven that l_1 and l_2 are parallel lines. It is already given that ∠ 1 is congruent to ∠ 2. ∠ 1 ≅ ∠ 2 The diagram shows that ∠ 2 and ∠ α are vertical angles. By the Vertical Angles Theorem, these angles are congruent. ∠ 2 ≅ ∠ α Notice the common angle of ∠ 2 in both relationships. By the Transitive Property of Congruence, since ∠ 1 is congruent to ∠ 2 and ∠ 2 is congruent to ∠ α, then ∠ 1 is congruent ∠ α. ∠ 1 ≅ ∠ 2 ∠ 2 ≅ ∠ α ⇓ ∠ 1 ≅ ∠ α The diagram also shows that ∠ 1 and ∠ α 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 ∠ 1 and ∠ α are corresponding congruent angles, then l_1 and l_2 are parallel lines. To summarize, all of the steps will be described in a two-column proof.

Statement	Reason
1. ∠ 1 ≅ ∠ 2	1. Given
2. ∠ 2 ≅ ∠ α	2. Vertical Angles Theorem
3. ∠ 1 ≅ ∠ α	3. Transitive Property of Congruence
4. l_1 ∥ l_2	4. Converse Corresponding Angles Theorem