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 $\angle RPS$

• A. ${86}^o$
• B. ${96}^o$
• C. ${92}^o$
• D. ${100}^o$

Answer 1

The correct option is B ${96}^o$

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 ]

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