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Alternate Exterior Angles: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Alternate Exterior AnglesAlternate Exterior Angles: Definition and ExamplesTable of ContentsAlternate Exterior Angles Definition of Alternate Exterior Angles

Alternate exterior angles are formed when a transversal line cuts across two other lines. These angles appear on opposite sides of the transversal and outside the two lines being intersected. When you look at the angles created, those that lie on alternate sides of the transversal and on the exterior sides of the two lines are called alternate exterior angles.

The Alternate Exterior Angles Theorem states that when two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent (equal in measure). The converse of this theorem is also true: if the alternate exterior angles formed by two lines cut by a transversal are congruent, then the two lines must be parallel to each other. This relationship helps us prove whether lines are parallel or find missing angle measures.

Examples of Alternate Exterior Angles Example 1: Finding an Unknown Angle Problem:

Find the value of xxx.

Alternate Exterior Angles

Step-by-step solution:

Step 1, Look at the given information. We know that angle EFHEFHEFH is 130°130°130°.

Step 2, Find the measure of angle GFHGFHGFH using the linear pair relationship. Since angles in a linear pair add up to 180°180°180°, we can write:

m∠EFH+m∠GFH=180°m∠EFH + m∠GFH = 180°m∠EFH+m∠GFH=180°

130°+m∠GFH=180°130° + m∠GFH = 180°130°+m∠GFH=180°

m∠GFH=180°−130°=50°m∠GFH = 180° - 130° = 50°m∠GFH=180°−130°=50°

Step 3, Use the Alternate Exterior Angles Theorem to find xxx. Since angle GFHGFHGFH and angle ACBACBACB (which equals xxx) are alternate exterior angles, they are equal:

m∠GFH=xm∠GFH = xm∠GFH=x 50°=x50° = x50°=x

Step 4, The answer is x=50°x = 50°x=50°.

Example 2: Determining When Lines are Parallel Problem:

If two alternate exterior angles formed by two lines and a transversal are (2x+10)(2x + 10)(2x+10) and (4x−20)(4x - 20)(4x−20), then what should the value of xxx be such that two lines are parallel?

two alternate exterior angles formed by two lines

Step-by-step solution:

Step 1, Remember that for two lines to be parallel, their alternate exterior angles must be equal.

Step 2, Set up an equation by making the two alternate exterior angles equal:

(2x+10)=(4x−20)(2x + 10) = (4x - 20)(2x+10)=(4x−20)

Step 3, Solve the equation for x:

2x+10=4x−202x + 10 = 4x - 202x+10=4x−20

10+20=4x−2x10 + 20 = 4x - 2x10+20=4x−2x

30=2x30 = 2x30=2x

x=302=15x = \frac{30}{2} = 15x=230​=15

Step 4, The value of xxx must be 151515 for the two lines to be parallel.

Example 3: Finding a Variable Value with Parallel Lines Problem:

Find the value of xxx if the two parallel lines are cut by a transversal.

Alternate Exterior Angles

Step-by-step solution:

Step 1, Identify the alternate exterior angles in the figure. They are given as (3x+49)(3x + 49)(3x+49) and (7x−55)(7x - 55)(7x−55).

Step 2, Apply the Alternate Exterior Angles Theorem. Since the lines are parallel, the alternate exterior angles must be equal:

(3x+49)=(7x−55)(3x + 49) = (7x - 55)(3x+49)=(7x−55)

Step 3, Solve the equation to find xxx:

3x+49=7x−553x + 49 = 7x - 553x+49=7x−55

49+55=7x−3x49 + 55 = 7x - 3x49+55=7x−3x

104=4x104 = 4x104=4x

x=26x = 26x=26

Step 4, The value of xxx is 262626.

Comments(2)SStreamerKimNovember 4, 2025I've used this alternate exterior angles def. with my students. It's super clear and the examples really helped them grasp the concept.

NNatureLover75September 17, 2025I’ve been struggling to explain alternate exterior angles to my kids, but this page broke it down so clearly! The examples made it easy for them to grasp, and now they’re solving problems confidently. Thank you!

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