In the parallelogram ABCD, the angles A and C are obtuse. Points X and Y are taken on the diagonal BD such that the angles XAD and YCB are right angles. Prove that : XA = YC.

Parallelogram ABCD is shown in the figure below:

We know that,

Opposite angles of a parallelogram are equal.

∴ ∠B = ∠D = z (let).

Diagonals bisect the interior angles in a parallelogram.

Thus, BD bisects angles ∠B and ∠D.

∴ ∠YBC = ∠ADX = $\dfrac{z}{2}$.

In △ XAD and △ YCB,

⇒ ∠XAD = ∠YCB (Both equal to 90°)

⇒ ∠ADX = ∠YBC (Both equal to $\dfrac{z}{2}$.)

⇒ AD = BC (Opposite sides of parallelogram are equal.)

∴ △ XAD ≅ △ YCB (By A.S.A. axiom).

We know that,

Corresponding parts of congruent triangles are equal.

∴ XA = YC.

Hence, proved that XA = YC.