Question

In the given figrue, ABCD is a cyclic quadrialteral. A circle passing through A and B meets AD and BC at the points P and Q respectively. Prove that PQ is parallel to DC.

Answer 1

Now ABQP is a cyclic quadrilateral
And we know opposite angles of a cyclic quadrilateral are supplementary
Hence $\angle x+\angle y={180}^{\circ }.......\left(1\right)$
Also ABCD is a cyclic quadrilateral
And we know opposite angles of a cyclic quadrilateral are supplementary
Hence $\angle x+\angle z={180}^{\circ }.......\left(2\right)$
Now from (1) and (2) we can say that
$\angle x+\angle y=\angle x+\angle z\Rightarrow \angle y+\angle z$
But these are corresponding angles
$\therefore PQ||DC$