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 if they have the same shape but not necessarily the same size. This means:
- All corresponding angles are equal in measure
- All corresponding sides are proportional (in the same ratio)
- One triangle is a scaled version of the other
Symbol: △ABC ~ △DEF (read as "triangle ABC is similar to triangle DEF")
Important: All congruent triangles are similar (same size means same shape), but not all similar triangles are congruent (same shape doesn't mean same size).
The Three Similarity Tests
1. AA (Angle-Angle)
If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. (The third angle is automatically equal since angles sum to 180°.)
2. SSS Similarity (Proportional Sides)
If all three pairs of corresponding sides are proportional (have the same ratio), then the triangles are similar.
3. SAS Similarity (Proportional Sides + Included Angle)
If two pairs of sides are proportional AND the included angles are equal, then the triangles are similar.
Special Case: Parallel Lines
When a line is drawn parallel to one side of a triangle, it creates a smaller triangle that is similar to the original. Parallel lines create equal corresponding angles, so by AA the triangles are similar.
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!
Conclusion
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