Question

ABCD is a parallelogram a circle through vertices A and B meets side BC at point P and side AD at point Q so that quadrilateral PCDQ is cyclic

Answer 1

Answer:

Given data:

ABCD is a parallelogram.

A circle is drawn in such a manner that it passes through the vertices A and B and meets sides BC at P and side AD at Q.

To show: PCDQ is a cyclic quadrilateral

Solution:

From the figure attached below, we can see that points A, B, Q & P lies inside the circle, so, ABQP is a cyclic quadrilateral.

∴ ∠A = ∠CPQ …… [∵ the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle] ….. (i)

Now, we know ABCD is a //gm,

∴ ∠A + ∠D = 180° ….. [∵ Consecutive angles of a //gm are supplementary] …… (ii)

Thus,

From (i) & (ii), we get

∠CPQ + ∠D = 180°

Since opposite angles in a cyclic quadrilateral are supplementary

∴ PCDQ is a cyclic quadrilateral

Hence proved