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 $k\parallel l$.