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

angle1 angle angle7

and

angle4 angle angle6.

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

Proof.

Since l || m, by the Corresponding Angles Postulate,

angle1 is congruent to angle5.

Also, by the Vertical Angles Theorem,

angle5 is congruent to angle7.

Then, by the Transitive Property of Congruence,

angle1 is congruent to angle7.

You can prove that angle4 and angle6 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.