A set of quadrangles with equal opposite sides and diagonal angles is not necessarily a parallelogram.

Reason: As shown in the figure, quadrilateral ABCD is a parallelogram, connecting AC and AE perpendicular to BC and E;

Intercept EC'=EC and connect AC' on EB, and then connect ⊿AEC'≌⊿AEC,AC'=AC.

Turn ⊿acd∠CAC' clockwise around point A, and AC and AC' coincide.

Obviously quadrilateral ABC'D' satisfies: AB = CD = C 'D∠B=∠D=∠D', and quadrilateral ABC'D' is not a parallelogram.