If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

This can be proven using transformations.
Consider the triangles $△ABC$ and $△DEF,$ where If one of these triangles can be mapped to the other using only similarity transformations, they are similar. As the only condition for segments to be congruent is to have the same length, it's possible to dilate $△DEF$ by some factor so that $\overline{AB}$ and the image of $\overline{DE}$ are congruent.

By the ASA congruence theorem, the triangles are congruent. Consequently, there exists a rigid motion that maps $△D^{\prime }{E}^{\prime }{F}^{\prime }$ onto $△ABC.$ Thus, it is possible to combine a dilation with some rigid motion, both being similarity transformations, to map $△DEF$ onto $△ABC$. This means that the original triangles indeed are similar.