Question

Prove that an isometry takes a (poylgon) unicorn to a congruent unicorn.

Answer 1

The common notion is that two triangles are congruent precisely when there exists an isometry of the plane mapping one triangle onto the other. Reflections are isometries (i.e., bijective distance preserving functions), so most certainly the two triangles above are congruent.

It is perfectly sensible to consider shapes to be congruent only if one can be mapped onto another by an orientation preserving isometry. In fact, one can start with any group of transformations of the plane and define the corresponding notion of 'congruent' shapes, with respect to the chosen group. It is up to you which group you choose. The common notion of congruence in the plane is for the group of all isometries. It is not trivial but not very hard to characterise the isometries of the plane. It then becomes a theorem (not a definition as some high school texts will teach you) that two triangles are congruent iff they have the same side lengths (or any one of another such characterisation).