Question

In the 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

Answer: (A)

$2\angle SOP$

Given $PQRS$ is a quadrilateral and $SO$ and $PO$ are bisectors $OS$ two adjacent sides meet at point $O$

Then $\angle S=2\angle OSP$ $\angle P=2\angle OPS$

Then $\angle P+\angle R+\angle Q+\angle R={360}^0$

$\Rightarrow 2\angle OSP+2\angle OPS+\angle Q+\angle R={360}^0$............(1)

$\Delta SOP$

$\angle OSP+\angle OPS+\angle SOP={180}^0$

$\Rightarrow \angle OSP+\angle OPS=180-\angle SOP$(2)

Asper (1) and (2) we get

$\therefore 2(180-\angle SOP)+\angle Q+\angle R=360$

$\Rightarrow \angle Q+\angle R=2\angle SOP$