Was reflection allowed in Euclid's Elements? I seem to remember my high school teacher in geometry saying that it was not allowed. Yet it seems to me that the side-angle-side rule would allow two reflected triangles to be considered to be congruent. The reason that this question concerns me is that I read that a renowned mathematician said that the proof of the pythagorean theorem that uses reflected right triangles (they are arranged in two squares of sides "a" + "b" in the proof), is invalid because reflected triangle are not considered the same, i.e. not congruent because one can't fit them together.