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Alternate Exterior Angles Theorem

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 $∠ 4 ≅ ∠ 6$ .

Two parallel lines cut by a transversal n, with angles labeled 1 through 8

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 $k ∥ l$ .