I’ve been feeling the need for a little geometry review, so today I read the first few chapters of the LibreTexts book Elementary College Geometry. I stopped when I got to the section on triangle congruence properties, though, as I still want to prove some of those myself. SAS is proposition 4 in Euclid’s Elements, which at least narrows down what tools are needed for the proof. (Some subset of the constructions in propositions 1-3.)
When you do read Euclid’s proof, you’ll find that in order to get himself started he does something there that he never does again in the Elements. I think maybe his perspective would be something like, “I REALLY don’t want to assume SAS, so I’ll let myself use a method of proof I really don’t like here. I wonder if there’s a way around this.” In more modern times, people might either do geometry differently by using this method more generally, or else fess up and admit that SAS is a postulate.
So look at I. 4 with interest, but don’t assume that Euclid is going to be completely scrupulous here, and that his proof has to be the way you just described it.
I hope that doesn’t take away a lovely day of exploration, but I don’t want you to go mad looking for what’s not there.
Thank you for the warning. I reckon I’m already mad enough without courting any new opportunities to go that way.