Identifying Similar Triangles

Overview

There are three shortcuts for proving two triangles similar without checking every angle and every side. The AA Postulate needs only two pairs of congruent angles. SSS Similarity needs all three pairs of sides proportional, and SAS Similarity needs two pairs of sides proportional with the included angles congruent. The work of this lesson is choosing the right test and verifying its condition completely — especially for SSS, where all three sides must be checked.

Vocabulary

AA Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
SSS Similarity Theorem: If all three sides of one triangle are proportional to all three sides of another triangle, then the triangles are similar.
SAS Similarity Theorem: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
Included Angle: The angle formed by two sides of a triangle; for SAS Similarity the congruent angle must be the one between the two proportional sides.

Main Ideas

AA: two congruent angle pairs are enough, because the third angles are then congruent automatically (the angles of a triangle add to 180°).
SSS Similarity: form all three ratios of corresponding sides — the triangles are similar only if all three ratios are equal.
SAS Similarity: the congruent angle must be the included angle, between the two pairs of proportional sides.
When a pair of triangles fails any one check, the triangles are not similar, even if the other checks pass.
Write the similarity statement in corresponding order once similarity is established.

Lesson Notes

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Lesson Video

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Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.