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Alternate Angles: Definition and Examples | EDU.COM

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

Alternate angles are non-adjacent angles that lie on opposite sides of the transversal when it cuts across two lines. These angles don't share any common vertices and are formed when a transversal intersects two lines (either parallel or non-parallel). The key characteristic of alternate angles is their positioning - they must be on opposite sides of the transversal.

Based on their position relative to the parallel lines, alternate angles are categorized into two types: alternate interior angles and alternate exterior angles. Alternate interior angles are pairs of angles that lie in the inner region between the two parallel lines but on opposite sides of the transversal. These are sometimes called "Z-angles" due to their Z-shaped arrangement. Alternate exterior angles, on the other hand, are angle pairs that lie in the outer region of the two parallel lines, again on opposite sides of the transversal.

Examples of Alternate Angles Example 1: Determining if Lines are Parallel Using Alternate Interior Angles Problem:

Use the alternate interior angles theorem to determine if the lines cut by the transversal are parallel.

Alternate Angles

Step-by-step solution:

Step 1, Find the value of angle AAA. Angle AAA and the angle measuring 60°60°60° form a straight angle.

m∠A+60°=180°m \angle A + 60° = 180°m∠A+60°=180°

m∠A=120°m \angle A = 120°m∠A=120°

Step 2, Find the value of angle BBB. Similarly, angle BBB and 120°120°120° form a straight angle, so:

m∠B+120°=180°m \angle B + 120° = 180°m∠B+120°=180°

m∠B=60°m \angle B = 60°m∠B=60°

Step 3, Compare the alternate interior angles. Angle AAA and the original 120°120°120° angle are alternate interior angles and are equal. Angle BBB and the original 60°60°60° angle are also equal alternate interior angles.

Step 4, Apply the alternate interior angles theorem. Since the alternate interior angles are equal, the lines cut by the transversal must be parallel.

Example 2: Finding Unknown Angles with Parallel Lines Problem:

In the diagram given below, the lines cut by the transversal are parallel. Determine the measures of the angles AAA, BBB, and CCC.

Alternate Angles

Step-by-step solution:

Step 1, Use the alternate interior angles theorem to find angle AAA. Angle AAA and 155°155°155° are alternate interior angles. Since the lines are parallel, alternate interior angles are equal.

∠A=155°\angle A = 155°∠A=155°

Step 2, Find the measure of angle BBB using the straight angle property. Angle AAA and angle BBB form a straight angle, so:

∠A+∠B=180°\angle A + \angle B = 180°∠A+∠B=180°

155°+∠B=180°155° + \angle B = 180°155°+∠B=180°

∠B=25°\angle B = 25°∠B=25°

Step 3, Find the measure of angle CCC using the alternate interior angles theorem. Since angle BBB and angle CCC are alternate interior angles and the lines are parallel, they are equal.

∠C=∠B=25°\angle C = \angle B = 25°∠C=∠B=25°

Step 4, State all angle measures. The measures of angles AAA, BBB and CCC are 155°155°155°, 25°25°25° and 25°25°25° respectively.

Example 3: Solving for an Unknown Variable with Alternate Exterior Angles Problem:

In the figure given below, CECECE is parallel to FHFHFH. Find the value of xxx.

Alternate Angles

Step-by-step solution:

Step 1, Identify the relevant angles. In the given figure, ∠ADE∠ADE∠ADE and ∠FGJ∠FGJ∠FGJ form a pair of alternate exterior angles.

Step 2, Apply the alternate exterior angles theorem. Since CECECE is parallel to FHFHFH, these alternate exterior angles are equal.

∠ADE=∠FGJ\angle ADE = \angle FGJ∠ADE=∠FGJ

Step 3, Set up an equation using the given measurements. The sum of x°x°x° and 50°50°50° equals 130°130°130°.

x°+50°=130°x° + 50° = 130°x°+50°=130°

Step 4, Solve for xxx.

x=130−50x = 130 - 50x=130−50

x=80x = 80x=80

Step 5, State the answer. The value of xxx is 808080.

Comments(2)DDadOf3BoysNovember 6, 2025I've used this alternate angles glossary page with my students. It's a great resource, making the concept easy to understand with examples.

MCMs. CarterSeptember 17, 2025This explanation of alternate angles was so clear and easy to follow! I used it to help my son with his homework, and he finally gets the concept. The examples were super helpful too—thank you!

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