If the two triangles shown were similar but not congruent then it would
be possible to move the triangle A'B'C', with at most one reflection about
an angle bisector, to achieve the following diagram, where A' has been
mapped onto A and the angles AC'B' and ACB are equal as are angles AB'C'
and ABC.
But this would mean that the angles of the quadrilateral C'CBB' summed
to 360 degrees, an impossibility. Hence the triangles are congruent.
