Triangle Similarity Tests

An interactive demonstration of the three main tests for triangle similarity: AA, SSS (proportional), and SAS (proportional).

Prerequisites: triangle-congruence · Difficulty: beginner

Two triangles are similar when they have the same shape but possibly different sizes: equal corresponding angles and proportional sides, written $△ABC\sim △DEF$.

Strategy: there are three tests for similarity — $AA$ (two equal angles), $SSS$ (three proportional sides), and $SAS$ (two proportional sides with the included angle equal). The animation walks through each, plus the parallel line case and how similarity compares with congruence.

What are Similar Triangles?

Similar triangles have the same shape but may be different sizes. Symbol: $△ABC\sim △DEF$ means "triangle ABC is similar to triangle DEF."

Test 1: AA (Angle-Angle)

If two angles of one triangle equal two angles of another, the triangles are similar. (The third angle is automatically equal since angles sum to 180°.)

Test 2: SSS Similarity (Proportional Sides)

If all three pairs of corresponding sides are proportional (same ratio), the triangles are similar.

Test 3: SAS Similarity (Proportional Sides + Included Angle)

If two pairs of sides are proportional AND the included angle is equal, the triangles are similar. The angle must be BETWEEN the two sides!

Special Case: Parallel Lines

When a line is parallel to one side of a triangle, it creates a smaller similar triangle inside. Parallel lines create equal corresponding angles!

Similarity vs Congruence

Similar triangles have the same shape (equal angles, proportional sides). Congruent triangles have the same shape AND same size. All congruent triangles are similar, but not all similar triangles are congruent!

Summary - The Three Similarity Tests

The three similarity tests:

• AA: Two angles equal (most common)
• SSS Similarity: Three sides proportional
• SAS Similarity: Two sides proportional + included angle equal

Similar = same shape. Congruent = same shape AND size!

Triangle similarity means two triangles have the same shape but possibly different sizes. You can prove similarity using:

• AA: Two angles equal (most common test)
• SSS Similarity: Three sides proportional (all ratios equal)
• SAS Similarity: Two sides proportional and included angle equal

Remember:

• Similar triangles have equal angles and proportional sides
• Congruent triangles are similar with a scale factor of 1
• Parallel lines often create similar triangles

Notes

Two triangles are similar if they have the same shape but not necessarily the same size: all corresponding angles are equal, all corresponding sides are proportional, and one triangle is a scaled version of the other.

Symbol: $△ABC\sim △DEF$ (read "triangle $ABC$ is similar to triangle $DEF$").

Important: all congruent triangles are similar (same size implies same shape), but not all similar triangles are congruent (same shape does not imply same size).

Special case — parallel lines. A line drawn parallel to one side of a triangle cuts off a smaller triangle similar to the original: the parallel creates equal corresponding angles, so the two triangles match by $AA$.