Alternate Exterior Angles

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When a transversal intersects two lines, it forms two pairs of alternate exterior angles. The alternate exterior angles are the opposing pair of exterior angles formed by the transversal and the two lines.

Alternate exterior angles definition

Alternate exterior angles are pairs of opposing angles that are formed when a transversal intersects two lines. A transversal is a line that passes through two lines in the same plane at two distinct points. In the diagram below, transversal l intersects lines m and n forming two pairs of alternate exterior angles: ∠1 and ∠4 and ∠2 and ∠3.

Alternate exterior angles theorem

The alternate exterior angles theorem states that if a transversal cuts two parallel lines, the pairs of exterior angles formed are congruent. In the figure below, transversal l intersects lines m and n forming 8 angles. The pairs of alternate exterior angles are ∠1 and ∠7 and ∠4 and ∠6. Since m and n are parallel, ∠1 and ∠4 are congruent and ∠2 and ∠3 are congruent.

Converse of the alternate exterior angles theorem

The converse of the alternate exterior angles theorem states that if the alternate exterior angles of two lines cut by a transversal are congruent, the two lines are parallel. Since the alternate exterior angles in the figure below are congruent, we know that lines m and n are parallel.

Example:

Find the measures of angles 1, 2, and 4 below given that lines m and n are parallel.

Since lines m and n are parallel, ∠2=60°. Since ∠1 and ∠2 form a straight angle, ∠1=180°-60°=120°. Similarly, since the angle measuring 60° adjacent to ∠4 form a straight angle, ∠4=120°.

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