I have a question about Hilbert's axiomatic system. In this system we have the SAS (Side-Angle-Side) congruence criterion as an axiom, and then we prove the SSS (Side-Side-Side) and ASA (Angle-Side-Angle) criteria as theorems. I was wondering then if it was possible to have SSS as an axiom (eventually adding something more) and prove SAS as a theorem, and then again with ASA instead. I have tried to do some research but I didn't find anything at all. Any reference or result would be welcome, thanks a lot in advance.
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Though not necessarily an answer, if you assume some level of measurement, SAS follows from SSS according to https://link.springer.com/article/10.1007/s00022-015-0264-4 – C. Caruvana Apr 17 '25 at 03:28