Which postulate can be used to prove that the triangles are congruent?

You have two angles congruent, plus a side that's NOT between them.
I guess you'd call that situation " AAS " for "angle-angle-side".
That's what you have, and it's NOT enough to prove the triangles
congruent.  There can be many many different pairs of triangles
that have AAS = AAS.

So there's no congruence postulate to cover this case, because they're
not necessarily.