geometry on quadrilaterals

Question

In a quad. ABCD with AB=CD, P and Q are mid points of diagonals AC and BD.P and Q joined and extended hits both sides AB and CD at S and T respectively. How can I prove that angle AST=angle DTS?

Answer 1

Construction:-

(1) Through D, draw DX // AB such that DX = AB.

(2) CX extended meets AB extended at Y.

[Note that:- P is NOT a point on DX.]

Hint:-

(1) Is ABXD a //gm?

(2) Will PQ // CX?

(3) Show that α = β = γ = δ = ε.