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 .