Is triangle congruence SAS an axiom? I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms.
Thank you for your help! 

So I actually spend a good bit of time trying to think through this a few years back. This doesn't fully answer the question the way you will want it to, but it won't fit in a comment.
Firstly since SAS is true, (and relatively simple so Godel is unlikely to show up here) it has to follow from Euclid's axioms (or Hamilton's but they are equivalent)
Secondly, using some trig (law of sines and cosines stuff) you can show that SAS and SS and AAS are all equivalent (as in if one holds all the rest do by CPCTC). Then you can use some coordinate geometry to show that SSS (which is SS) must work.
You can do this by noting that rotations are isometries (preserve distance) and so do translations. So then you move the triangles such that they overlap at all points and then they must be congruent because every part of them overlaps, the sides must be the same length and so do the angles.
This isn't entirely rigorous, but it made me personally feel a bit better about the various triangle proofs. It is entirely possible that full rigor for something like this is very complex.
Hope the above is interesting/useful even if not entirely complete. I could go back and look for my notes on how I did the specifics of the above, but I think it was mostly trig and coordinate geometry reasoning. 
