3. In order to develop trigonometry, one needs the fact that there are 180o in a triangle, and that triangles which have the same angles are similar. We have been able to prove that there are 180o 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.

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Analytic Foundations of Geometry

R. S. Wilson