Question

In the figure, O is the centre of the circle. If $\angle BOD={160}^{\circ }$, find the values of x and y.

Answer 1

In the figure, O is the centre of the circle. ABCD is a cyclic quadrilateral.

$\angle BOD={160}^{\circ }$

$\because$ arc BAD subtends $\angle BOD$ at the centre and $\angle BCD$ on the other part of the circle.

$\therefore \angle BCD=\frac{1}{2}\angle BOD$

$\Rightarrow x=\frac{1}{2}\times {160}^{\circ }={80}^{\circ }$

$\because$ ABCD is a cyclic quadrilateral,

$\therefore \angle A+\angle C={180}^{\circ }\Rightarrow y+x={180}^{\circ }$

$\Rightarrow y+{80}^{\circ }={180}^{\circ }\Rightarrow y={180}^{\circ }-{80}^{\circ }={100}^{\circ }$

$\therefore x={80}^{\circ },y={100}^{\circ }$