I was looking at Euclid's proof of the triangle Side-Angle-Side equality, and I'm not really convinced by it.
The first step of the proof is to place the first triangle onto the second one, but why can we do this? I agree that we are able to translate lines $DE$ and $DF$ on top of $AB$ and $AC$, but we are only able to place $EF$ on top of $BC$ if their lengths are the same, so saying that we can just "place one triangle on top of the other" seems to be assuming the SAS equality.
For example, what if instead of the Euclidean plane, we took a plane but with a bump in between $B$ and $C$ such that the length of $BC$ is longer than $EF$ as it has to go over the bump. The triangles would have an equal angle between two equal sides, but we couldn't just place $ABC$ on top of $DEF$ as it wouldn't lie flat on the plane.
Is there a hidden assumption that we are in fact able to translate and rotate any triangle without changing its lengths that's missing from Euclid's list of axioms?