ABCD is a quadrilateral with AB equal and parallel to DC. Prove that AD is equal and parallel to BC

Question

This question is based on knowledge from vectors

Answer 1

Join A and C (diagonal). In triangles ABC and ACD ∠BAC=∠ACD (alternate angles between parallels AB and DC); AC is common; AB=DC (given). So the triangles are congruent (SAS, two sides and included angle). Therefore ∠DAC=∠ACB, and DA=CB in length. But these are alternate angles (opposite sides of the diagonal), so DA is parallel to CB.