PRINCIPIA · THEOREM

AA similarity

Dependencies: Triangle interior angles sum to 180°, SAS congruence, SAS similarity axiom.

Statement

Suppose △ABC\triangle ABC and △A′B′C′\triangle A'B'C' are both nondegenerate, and satisfy

∠A=∠A′,∠B=∠B′.\angle A = \angle A',\qquad \angle B = \angle B'.

Then the two triangles are similar:

△ABC  ∼  △A′B′C′.\triangle ABC \;\sim\; \triangle A'B'C'.

That is, as soon as two pairs of corresponding angles are equal, the triangles must be similar — the three pairs of sides are automatically in the same ratio, and the third pair of angles is automatically equal.

AA similarity: \angle A = \angle A' and \angle B = \angle B' ⇒ \triangle ABC \sim \triangle A'B'C'

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