Does the diagonal of a parallelogram bisect its angles?

Yes, we will prove it.
Given:
ABCD – parallelogram,
AC and BD are diagonals.
Prove that the diagonal of a parallelogram bisects its angles.
Evidence:
1) Consider triangles ABC and ACD. They have AB = CD and BC = AD because ABCD is a parallelogram, the AC side is common. Therefore triangle ABC = triangle CDA on three sides. Then the angle BCA = angle DCA;
2) Consider triangles BCD and ABD. They have AB = CD and BC = AD because ABCD is a parallelogram, side BD is common. Therefore triangle ABD = triangle BCD on three sides. Then the angle ABD = angle DVS.
Q.E.D.