Question

In the given figure, SO and PO are bisectors of two adjacent sides of quadrilateral, $\angle$Q + $\angle$R is:

• A. 2$\angle SOP$
• B. $\angle OSP$ + $\angle OPS$
• C. $\angle SOP$
• D. 2($\angle OSP$ + $\angle OPS$)

Answer 1

The correct option is A 2$\angle SOP$

From the figure,

$\angle P+\angle Q+\angle R+\angle S=360$

$⟹\angle Q+\angle R=360-(\angle P+\angle Q)⟶\left(1\right)$

In $△POS$

$\angle POS+\frac{1}{2}\angle P+\frac{1}{2}\angle S=180°$

On multiplying by 2on both sides

$2\angle POS+\angle P+\angle Q=360°$

$\angle P+\angle Q=360-2\angle AOB⟶\left(2\right)$

On substituting values of $\left(2\right)$ in $\left(1\right)$

$\angle Q+\angle R=360-(360-2\angle POS$

$⟹\angle Q+\angle R=2\angle POS$

$⟹\angle Q+\angle R=2\angle SOP$