In elliptic and hyperbolic geometry, I believe an equilateral triangle is uniquely determined up to congruence by one of its angles, but I'm not sure of the proof.
If the conditions are equivalent for triangles, are there two words because they are also used to describe equilateral / equiangular polygons, which are not always the same? For example, a rectangle is equiangular but need not be equilateral, and a rhombus is equilateral but need not be equiangular. What's so special about triangles? (or tetrahedra?)