Criteria for Triangle Similarity

AAA, SSS, SAS, and AA Similarity Tests

"Similar triangles have the same shape but different sizes - their corresponding angles are equal and corresponding sides are proportional"
AAA/AA Similarity
∠A = ∠D, ∠B = ∠E, ∠C = ∠F
All corresponding angles equal
SSS Similarity
AB/DE = BC/EF = CA/FD
All sides proportional
SAS Similarity
AB/DE = AC/DF, ∠A = ∠D
Two sides proportional, included angle equal
Welcome to the world of triangle similarity! Let's discover the powerful criteria that determine when triangles are similar.
🎯 What is Triangle Similarity?

Two triangles are similar if they have the same shape but not necessarily the same size. This means all corresponding angles are equal and all corresponding sides are proportional.

ΔABC ~ ΔDEF
Triangle ABC is similar to triangle DEF
📐 Angle-Angle-Angle (AAA)

If all three corresponding angles of two triangles are equal, then the triangles are similar. Since the sum of angles in a triangle is 180°, knowing two angles determines the third!

∠A = ∠D, ∠B = ∠E ⟹ △ABC ~ △DEF
Two angles equal suffice (AA criterion)
📏 Side-Side-Side (SSS)

If the ratios of all three corresponding sides of two triangles are equal, then the triangles are similar. This ensures the same shape with proportional scaling.

AB/DE = BC/EF = CA/FD = k
All sides in same ratio k
🔄 Side-Angle-Side (SAS)

If two corresponding sides of two triangles are proportional and the included angles are equal, then the triangles are similar.

AB/DE = AC/DF, ∠A = ∠D
Two sides proportional + included angle equal

🎮 Interactive Similarity Explorer

Triangle Controls:

Similarity Analysis:

Scale Factor: 1.00
Angle A₁: 60° | Angle A₂: 60°
Angle B₁: 60° | Angle B₂: 60°
Angle C₁: 60° | Angle C₂: 60°
Similar? Yes