Proving that two triangles are similar is not too much of a hassle, since all you need to do is identify two common angles (or SAS/SSS). However, it seems that as soon as you move beyond the triangle, it becomes a whole lot harder. Take for instance two parallelograms. Whilst both of them might have the same corresponding angles, this does not guarantee that they are similar, for you can stretch one of the out so as to make the proportions of sides different, without affecting the angles.
My question refers to similarity for 3D shapes. Is there a clear way for determining whether two such shapes are similar. The only thing I managed to come up with is proving that each two corresponding faces of the two shapes are similar. Is this correct? I tried looking at two shapes, the tetrahedron (4 triangular faces) and the parallelepiped (3D parallelogram), but didn't manage to come up with a convincing argument.