What additional information is needed to prove the triangles are congruent by the sas postulate?

you need to know that ∠xwy = ∠zwy.

xw = wz because it's given, and we know that wy = wy (a side shared by both triangles) because of the identity property of equality. so to prove that both triangles are congruent with sas we need to know that those two angles are congrunt.