Triangles with Two Equal Angles are Similar

Theorem

Two triangles which have two corresponding angles which are equal are similar.

Proof

Let $\triangle ABC$ and $\triangle DEF$ be triangles such that $\angle ABC = \angle DEF$ and $\angle BAC = \angle EDF$.

Then from Sum of Angles of Triangle Equals Two Right Angles $\angle ACB$ is equal to two right angles minus $\angle ABC + \angle BAC$.

Also from Sum of Angles of Triangle Equals Two Right Angles $\angle DFE$ is equal to two right angles minus $\angle DEF + \angle EDF$.

That is, $\angle DFE$ is equal to two right angles minus $\angle ABC + \angle BAC$.

So $\angle DFE = \angle ACB$ and so all three corresponding angles of $\triangle ABC$ and $\triangle DEF$ are equal.

The result follows from Equiangular Triangles are Similar.

$\blacksquare$

Sources

• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.8$: Corollary $1$