Question

PQRS is a parallelogram whose diagonals intersect at M.
If $\angle PMS={54}^{\circ }$, $\angle QSR={25}^{\circ }$ and $\angle SQR={30}^{\circ }$: find $\angle PRS$

• A. ${39}^o$
• B. ${29}^o$
• C. ${18}^o$
• D. ${24}^o$

Answer 1

Answer: (B)

${29}^o$

Given is a parallelogram PQRS whose diagonals intersect at M.

Also given $\angle PMS={54}^o$, $\angle QSR={25}^o$ and$\angle SQR={30}^o$

We know that in a parallelogram opposite sides are parallel.

So PQ$\parallel$SR & if SQ is considered as the transversal then

$\angle PSM=\angle RQM={30}^o$[ Alternate Angles ]

Also $\angle PSR=\angle PSQ+\angle QSR={30}^o+{25}^o={55}^o$

Now in

$△PMS,\angle MPS+\angle PSM+PMS={180}^o\Rightarrow \angle MPS+{30}^o+{54}^o={180}^o\Rightarrow \angle MPS+{84}^o={180}^o\Rightarrow \angle MPS={180}^o-{84}^o\Rightarrow \angle MPS={96}^o$

Now $\angle RPS=\angle MPS={96}^o$

Now in

$△PSR,\angle PSR+\angle PRS+RPS={180}^o\Rightarrow \angle PRS+{96}^o+{55}^o={180}^o\Rightarrow \angle PRS+{151}^o={180}^o\Rightarrow \angle PRS={180}^o-{151}^o\Rightarrow \angle PRS={29}^o$