Question

PQRS is a parallelogram whose diagonals intersect at M.
$\angle PMS={54}^{\circ },\angle QSR={25}^{\circ }\text{and}\angle SQR={30}^{\circ };$
$\text{(i)}\angle RPS$
$\text{(ii)}\angle PRS$
$\text{(iii)}\angle PSR$

Answer 1

Given: Parallelogram PQRS in which diagonals PR and QS intersect at M.
$\angle PMS={54}^{\circ };\angle QSR={25}^{\circ }\text{and}\angle SQR={30}^{\circ }$


$\text{To find: (i)}\angle RPS\text{(ii)}\angle PRS\text{(iii)}\angle PSR$
$\Rightarrow >PSQ=\angle SQR\text{(Alternate}\angle S)$
$\text{But}\angle SQR={30}^{\circ }$
$\angle PSQ={30}^{\circ }$
$\text{In}△SMP,$
$\angle PMS+\angle PSM+\angle MPS={180}^{\circ }\text{or}{54}^{\circ }+{30}^{\circ }+\angle RPS={180}^{\circ }$
$\angle RPS={180}^{\circ }-{84}^{\circ }={96}^{\circ }$
$\text{Now,}\angle PRS+\angle RSQ=\angle PMS$
$\angle PRS+{25}^{\circ }={54}^{\circ }$
$\angle PRS={54}^{\circ }-{25}^{\circ }={29}^{\circ }$
$\angle PSR=\angle PSQ+\angle RSQ={30}^{\circ }+{25}^{\circ }={55}^{\circ }$
$\text{Hence, (i)}\angle RPS={96}^{\circ }\text{(ii)}\angle PRS={29}^{\circ }\text{(iii)}\angle PSR={55}^{\circ }$