Similarity Tests for Triangles (AA, SAS, SSS)
Simple Explanation
You do not need to check all three angles and all three sides to prove triangles are similar. Three shortcuts suffice: AA (two pairs of equal angles), SAS (two pairs of proportional sides with the included angle equal), or SSS (all three pairs of sides proportional).
Why Do We Need It?
These shortcuts make proving similarity practical — checking just two angles (AA) is far faster than measuring every side and angle of both triangles.
See It
Two triangles of different sizes with matching angle marks at two corresponding vertices, illustrating the AA similarity test
Formula
Similarity Ratio
AB/A'B' = BC/B'C' = CA/C'A' = k, and corresponding angles are equal
Two triangles are similar exactly when their corresponding angles are equal and their corresponding sides are all in the same fixed ratio, k (the scale factor).
- k
- — the scale factor — how many times larger (or smaller) one triangle is than the other
- AB, BC, CA / A'B', B'C', C'A'
- — corresponding side lengths of the two similar triangles
When to use it: Whenever you need to confirm two triangles are similar, or use their known similarity to find a missing side length.
Worked Example
Prove two triangles similar using AA
In △ABC and △DEF, ∠A = ∠D = 50° and ∠B = ∠E = 70°. Are the triangles similar?
Why Does This Work?
AA is enough because a triangle's three angles always sum to 180° — once two angles match, the third is forced to match too, so all three angle pairs are automatically equal, which is the defining condition for similarity. SAS and SSS work because, together with the triangle's rigidity, matching those specific combinations of proportional sides and angles is enough to force every other angle and side ratio to match as well.
Real-Life Example
Measuring height using shadows
A person's height and shadow length form a triangle similar to a tree's height and shadow, since both make the same angle with the sun's rays (AA — the right angle with the ground, and the shared sun angle).
Surveyors and students use exactly this AA similarity to measure a tall object's height indirectly, using only shadow lengths and a known height.
Practice
Which similarity test needs only two pieces of angle information?
MediumCommon mistake
Checking only one pair of equal angles and assuming that is enough — a single equal angle does not guarantee similarity; AA specifically needs two independent equal angle pairs.
Quick Review
- AA: two pairs of equal corresponding angles.
- SAS: two pairs of proportional sides with the included angle equal.
- SSS: all three pairs of corresponding sides proportional.