Maybe this link can help, though it does not prove the congruence of the triangles from scratch but rather reduces it to SAS. In turn, SAS is discussed in Proposition 4, but ultimately it is taken as an axiom.
From my recollection, proofs of congruence for triangles are the worst in some sense because they are so close to axioms that it is hard to distinguish something that has to be proved from something taken on faith. Therefore, don't worry if you can't always tell those things apart. Theorems that are studied later may have more complex proofs, but those proofs are more convincing.