geometry proof

Question

prove that the opposite sides of any parallelogram in euclidean geometry are congruent.

Answer 1

ABCD is a parallelogram.

In triangles ABD and BCD, BD, the diagonal of ABCD, is common to both triangles.

Angles BDA and DBC are equal (alternate angles between parallel lines)

Angles ABD and BDC are equal for the same reason.

The triangles are congruent (ASA) therefore BC=AD and AB=DC, the opposite sides of ABCD are equal (congruent).