# 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 $k\parallel l$ , then

$\angle 1\cong \angle 7$ and $\angle 4\cong \angle 6$ .

**
Proof.
**

Since $k\parallel l$ , by the Corresponding Angles Postulate ,

$\angle 1\cong \angle 5$ .

Also, by the Vertical Angles Theorem,

$\angle 5\cong \angle 7$ .

Then, by the Transitive Property of Congruence,

$\angle 1\cong \angle 7$ .

You can prove that $\angle 4$ and $\angle 6$ 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 $k\parallel l$ .