In the adjoining figure, ABCD is a parallelogram. Line segments AX and CY bisect ∠A and ∠C respectively. Prove that :

(i) ΔADX ≅ ΔCBY

(ii) AX = CY

(iii) AX ∥ CY

(iv) AYCX is a parallelogram

In the given figure, ABCD is a parallelogram and X, Y are points on diagonal BD such that DX = BY. Prove that CXAY is a parallelogram.

If a diagonal of a parallelogram bisects one of the angles of the parallelogram, prove that it also bisects the second angle and then the two diagonals are perpendicular to each other.

In the given figure, ABCD is a parallelogram and E is the mid-point of BC. If DE and AB produced meet at F, prove that AF = 2AB.