Triangles with Two Equal Angles are Similar
Let $\triangle ABC$ and $\triangle DEF$ be triangles such that $\angle ABC = \angle DEF$ and $\angle BAC = \angle EDF$.
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.