Question

In the given figure, ABCD is a quadrilateral and AO and DO are bisectors of $\angle Aand\angle D$. Find the value of x.

Answer 1

In a quadrilateral ABCD, AO and DO are bisectors of $\angle A\text{and}\angle D$.
So, $\angle DAO=\frac{\angle A}{2}\text{and}\angle ADO=\frac{\angle D}{2}$
$\because \text{Sum of angles in a quadrilateral is}{360}^0$
$\Rightarrow \angle A+\angle B+\angle C+\angle D={360}^{\circ }$
$\Rightarrow \angle A+\angle B+120+70={360}^{\circ }$
$\Rightarrow \angle A+\angle B={170}^{\circ }$
$\Rightarrow \frac{\angle A}{2}+\frac{\angle B}{2}={85}^{\circ }$ ---------(1)

$\because \text{Sum of angles in a triangle is}{180}^0$
$\Rightarrow \frac{\angle A}{2}+\frac{\angle B}{2}+x={180}^0$
$\Rightarrow x={180}^{\circ }-(\frac{\angle A}{\angle 2}+\frac{\angle B}{\angle 2})$
$\Rightarrow x={180}^{\circ }-{85}^{\circ }$
$={95}^{\circ }$
Hence, the value of '$x$' is ${95}^{\circ }$.