3. In order to develop trigonometry, one needs the fact that there
are 180^{o} in a triangle, and that triangles which have the
same angles are similar. We have been able to prove that there are
180^{o} in a triangle, but we have not yet been able to prove
that two triangles which have the same angles are similar. It is
possible to prove the similarity criterion for triangles without
using congruent triangles, but at this point it will take all of the
work which is involved with proving this theorem.

Analytic Foundations of Geometry