Question
In the given figure :if $\angle PQS = 60^\circ ,$find$\angle QPR.$
✓
Answer
In $\triangle PQS,$
$PQ = QS ....($given$)$
$\Rightarrow \angle QSP = \angle QPS ....($angles opposite to two equal sides are equal$)$
Now, $\angle PQS + QPS + \angle QPS = 180^\circ $
$\Rightarrow 60^\circ + 2\angle QSP = 180^\circ $
$\Rightarrow 2\angle QSP = 120^\circ $
$\Rightarrow \angle QSP = 60^\circ $
$\Rightarrow \angle QPS = 60^\circ $
In $\triangle PRS,$
$PS = SR ....($given$)$
$\Rightarrow \angle PRS = \angle RPS ....($angles opposite to two equal sides are equal$)$
By exterior angle property,
$\angle QSP = \angle RPS + \angle PRS$
$\Rightarrow 60^\circ = 2\angle RPS$
$\Rightarrow \angle RPS = 30^\circ $
Now,
$\angle QPR = \angle QPS + \angle RPS$
$= 60^\circ + 30^\circ = 90^\circ .$
$PQ = QS ....($given$)$
$\Rightarrow \angle QSP = \angle QPS ....($angles opposite to two equal sides are equal$)$
Now, $\angle PQS + QPS + \angle QPS = 180^\circ $
$\Rightarrow 60^\circ + 2\angle QSP = 180^\circ $
$\Rightarrow 2\angle QSP = 120^\circ $
$\Rightarrow \angle QSP = 60^\circ $
$\Rightarrow \angle QPS = 60^\circ $
In $\triangle PRS,$
$PS = SR ....($given$)$
$\Rightarrow \angle PRS = \angle RPS ....($angles opposite to two equal sides are equal$)$
By exterior angle property,
$\angle QSP = \angle RPS + \angle PRS$
$\Rightarrow 60^\circ = 2\angle RPS$
$\Rightarrow \angle RPS = 30^\circ $
Now,
$\angle QPR = \angle QPS + \angle RPS$
$= 60^\circ + 30^\circ = 90^\circ .$
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