Question

Find the measure of $\angle P$ and $\angle S$ if SP || RQ. (If you find m$\angle R$ , is there more than one method to find m$\angle P$ ?

Answer 1

$\text{Here,}$

$\angle P+\angle Q={180}^{\circ }\left[\text{Sum~of~co-interior~ angles~is~}{180}^{\circ }\text{as}SP||RQ\right]$
$\Rightarrow \angle P+{130}^{\circ }={180}^{\circ }$
$\Rightarrow \angle P={180}^{\circ }-{130}^{\circ }$
$\Rightarrow \angle P={50}^{\circ }$
$\angle R={90}^{\circ }\left[\text{Given}\right]$
$\angle S+{90}^{\circ }={180}^{\circ }[\text{Sum~of~co-interior~ angles~is~}{180}^{\circ }$
$\Rightarrow \angle S={180}^{\circ }-{90}^{\circ }$
$\Rightarrow \angle S={90}^{\circ }$
$\text{Yes,~we~can~use~another~method~to~find~}\angle P$
$\angle S+\angle R+\angle Q+\angle P={360}^{\circ }\left[\text{Angle~ sum~property~of~quadrilateral}\right]$
$\Rightarrow {90}^{\circ }+{90}^{\circ }+{130}^{\circ }+\angle P={360}^{\circ }$
$\Rightarrow {310}^{\circ }+\angle P={360}^{\circ }$
$\Rightarrow \angle P={360}^{\circ }-{310}^{\circ }$
$\Rightarrow \angle P={50}^{\circ }$