Understanding the Properties of Alternate Exterior Angles
POSTED ON MARCH 09, 2023

Knowing how to identify alternate exterior angles and applying their properties comes in handy when working with unknown angles of parallel lines. Master this topic and you’ll have one more property to add to your geometric toolkit.

## What Are Alternate Exterior Angles?

Alternate exterior angles are pairs of non-adjacent angles found at the outer side of the regions formed when two parallel lines are intersected by a transversal line. Each pair is located at alternating positions as shown below.

When parallel lines, m and n, are cut by the transversal line, eight angles are formed. For alternate exterior angles, we focus on the outer regions where angles ∠1, ∠2, ∠7, and ∠8 lies. These are the exterior angles and if we pair the angles lying on alternating positions, we’ll have the following pairs of alternate exterior angles: ∠1 and ∠8 as well as ∠2 and ∠7.

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Now that you know how to find pairs of alternate exterior angles, try out the problem on your own to test your understanding!

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#### Example 1

Two parallel lines, u and v, are cut by a transversal line as shown above. Which of the following pairs of angles are alternate exteriors angles?

A. ∠a and ∠d

B. ∠c and ∠g

C. ∠a and ∠h

D. ∠b and ∠h

Exterior angles are found at the outer sides of the transversal line, so that leaves us with the following angles: ∠a, ∠b, ∠g, and ∠h. This eliminates A and B.

Alternate exterior angles are found in alternate positions (hence, the name), so angles ∠a and ∠h are alternate exterior angles. This makes C the correct answer.

## Alternate Exterior Angles Theorem

Alternate exterior angles exhibit a special property: each pair of alternate exterior angles are congruent to each other. This property is known as the alternate exterior angles theorem.

Going back to the alternate interior angles, ∠1 & ∠8 and ∠2 & ∠7, each pair will have equal angle measures.

Remember this important property of alternate exterior angles because they will come in handy when finding unknown angles of parallel lines.

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#### Example 2

Use the alternate exterior angles theorem to find the measure of the following:

a. ∠g

By inspecting the parallel lines, s and t, we can see that the angles, ∠b and ∠g, are alternate exterior angles. This means that they have equal angle measures.

Since ∠b = 60°, ∠g will also measure 60°.

b. ∠h

Angles ∠a and ∠b form a line, so they will add up to 180°. This means that ∠a is equal to 180° - 60° = 120°. Now, ∠a and ∠h are alternate exterior angles, so they are congruent.

This means that ∠h has an angle measure of 120°.

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