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I am asked to prove the SAS (side-angle-side) and ASA(angle-side-angle) congruence of triangles in absolute geometry ( the geometry based in the axiom system of Euclides with the parallel postulate removed).

How do I even prove two triangles are congruent without using the SAS, ASA or SSS criteria? Do I have to see the congruence between any of sides or angles of the two triangles?

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  • $\begingroup$ This depends on exactly what your axioms are. For instance, if you are using Hilbert's axioms, then SAS itself is one of the axioms. $\endgroup$ – Eric Wofsey Oct 30 '17 at 20:58
  • $\begingroup$ @EricWofsey Well, not exactly one of the axioms, but nearly so. $\endgroup$ – Aretino Oct 30 '17 at 21:06

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