Proving Triangles are Similar

Similarity Tests (Shortcuts)

In geometry, when we talk about similarity shortcuts, we’re referring to the specific sets of conditions that allow us to quickly prove that two triangles are similar without having to verify that all three pairs of corresponding angles are equal AND all three pairs of corresponding sides are proportional.

These shortcuts leverage the inherent properties of triangles to simplify the proof process. There are three primary similarity shortcuts for triangles: AA (Angle-Angle) Similarity, SSS (Side-Side-Side) Similarity, and SAS (Side-Angle-Side) Similarity.

The following diagrams show how to use the AA Similarity Test or shortcut to determine whether two triangles are similar. Scroll down the page for more examples and solutions.

 

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1. AA (Angle-Angle) Similarity

• If two angles of one triangle are congruent (equal in measure) to two angles of another triangle, then the two triangles are similar.
• Why it’s a shortcut: Because the sum of the angles in any triangle is always 180°, if you know two pairs of corresponding angles are equal, the third pair must also be equal. Once all corresponding angles are equal, the sides are necessarily proportional, making the triangles similar.
• When to use it: This is often the easiest and most common shortcut to use. Look for parallel lines (creating corresponding or alternate interior angles), vertical angles, or shared angles in overlapping triangles.

2. SSS~ (Side-Side-Side) Similarity

• If the corresponding side lengths of two triangles are proportional (i.e., their ratios are equal), then the triangles are similar.
• Why it’s a shortcut: If all three sides are consistently scaled versions of each other, the angles between those sides must naturally match up to maintain the shape.
• When to use it: Use this when you know the lengths of all three sides for both triangles. You’ll calculate the ratio of the shortest side to the shortest side, the middle to the middle, and the longest to the longest. If all ratios are the same, they’re similar.

3. SAS~ (Side-Angle-Side) Similarity

• If two sides of one triangle are proportional to two corresponding sides of another triangle, AND the included angle (the angle between those two proportional sides) is congruent, then the triangles are similar.
• Why it’s a shortcut: The fixed angle between two proportionally scaled sides forces the third side to also be in proportion, completing the similar shape.
• When to use it: Use this when you know two sides and the angle between them for both triangles. You’ll check the proportionality of the sides and the equality of the included angle.

Similarity Shortcut
Triangle Similarity
There are four triangle congruence shortcuts: SSS, SAS, ASA, and AAS.
There are three triangle similarity shortcuts: AA, SAS~, SSS~
We have triangle similarity if
(1) two pairs of angles are congruent (AA)
(2) two pairs of sides are proportional and the included angles are congruent (SAS~), or
(3) if three pairs of sides are proportional (SSS~).

• Show Video Solutions

Investigate Properties of Similar Triangles Part 1
This lesson shows how to determine when triangles are congruent or similar

• Show Step-by-step Solutions

Investigate Properties of Similar Triangles Part 2
This lesson shows how to determine when triangles are congruent or similar.

• Show Video Solutions

Congruent and Similar Triangles
Working with similar triangles, determining similar triangles

• Show Step-by-step Solutions

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