T4. This is the start of the congruent triangle criteria. They are derived in trigonometry courses using the Law of Sines and the Law of Cosines, so one wonders if it might be easier to derive them analytically without having to use the traditional method of moving one triangle until it coincides with another using translations, rotations, and reflections, as we do here.

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