Given that ABCD is a parallelogram:
Opposite sides of a parallelogram are parallel and congruent. Therefore, AB = DC.
Diagonals of a parallelogram bisect each other. Therefore, the midpoint of AC is the same as the midpoint of BD. Let M be the midpoint of AC, and N be the midpoint of BD.
By the midpoint theorem, BM = DM and BN = AN.
Since BM = DM and BN = AN, we can conclude that quadrilateral ABCD is a parallelogram in which BC || AD and CD || AB.
Therefore, we have shown that AB = CD and BC = AD in parallelogram ABCD.