AA similarity

Theorem: In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.

Proof

Let ABC and DEF be two triangles such that $\angle A = \angle D$ and $\angle B = \angle E$. $\angle A + \angle B + \angle C = 180$ and $\angle D + \angle E + \angle F = 180$ Thus, we can write the equation: $\angle A  + \angle B + \angle C=\angle D + \angle E + \angle F \Longrightarrow \angle D + \angle E + \angle C = \angle D + \angle E + \angle F$, since we know that $\angle A = \angle D$ and $\angle B = \angle E$, from before. Therefore, by subtracting $\angle D + \angle E$ by both equations, we get $\angle C = \angle F$.


See also

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