Question

One side of a cyclic quadrilateral passes through the center of a circle. A pair of opposite sides of the quadrilateral is parallel. Determine the angle $\angle XWZ.$

• A. ${65}^o$
• B. ${100}^o$
• C. ${115}^o$

Answer 1

The correct option is C ${115}^o$

We know: A cyclic quadrilaterls has its all four points on circle.

One side of the cyclic quadrilateral passes through the center of the circle.
Hence, it is the diameter of the circle.

The angle subtended by the diameter to any point on the circle is equal to ${90}^o.$
$\therefore \angle XZY={90}^o$

Also, possible pair of opposite side is $XY\&WZ$.
Using transveral line rule,
$\therefore \angle YXZ$ and $\angle WZX$ are alternate interior angles of the transversal XZ.
$\Rightarrow \angle YXZ=\angle WZX={25}^o$

$\therefore \angle XYZ={180}^o-\angle YXZ-\angle YZX$
$\Rightarrow \angle XYZ={180}^o-{25}^o-{90}^o={65}^o$

XYZW is a inscribed quadrilateral.
The sum of opposite angles of a quadrilateral inscribed in a circle is ${180}^o.$
$\therefore \angle XWZ+\angle XYZ={180}^o$
$\Rightarrow \angle XWZ={180}^o-\angle XYZ={180}^o-{65}^o={115}^o$