AA similarity

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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 $\angle B = \angle E$. The sum interior angles of a triangle is equal to 180. Therefore, $\angle A + \angle B + \angle C = 180$, and $\angle D + \angle E + \angle F = 180$. We can write the equation: $\angle A  + \angle B + \angle C = 180 = \angle D + \angle E + \angle F \Longrightarrow \angle D + \angle E + \angle C = \angle D + \angle E + \angle F$, acknowledging the fact that $\angle A = \angle D$ and $\angle B = \angle E$. To conclude, by subtracting $\angle D + \angle E$ by both equations, we get $\angle C = \angle F$.


See also

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