What theorem which states that if two parallel lines are cut by a transversal then alternate exterior angles are congruent?

The Alternate Exterior Angles Theorem states that, when two parallel lines are cut by a transversal , the resulting alternate exterior angles are congruent .

So, in the figure below, if k ∥ l , then

∠ 1 ≅ ∠ 7 and ∠ 4 ≅ ∠ 6 .

Proof.

Since k ∥ l , by the Corresponding Angles Postulate ,

∠ 1 ≅ ∠ 5 .

Also, by the Vertical Angles Theorem,

∠ 5 ≅ ∠ 7 .

Then, by the Transitive Property of Congruence,

∠ 1 ≅ ∠ 7 .

You can prove that ∠ 4 and ∠ 6 are congruent using the same method.

The converse of this theorem is also true; that is, if two lines k and l are cut by a transversal so that the alternate exterior angles are congruent, then k ∥ l .