Question
In the given figure, $\angle PQR =\angle PRQ$, then prove that $\angle PQS =\angle PRT$
✓
Answer
We need to prove that $\angle PQS =\angle PRT$
We are given that $\angle PQR =\angle PRQ$
From the given figure, we can conclude that $\angle PQS$ and $\angle PQR$, and $\angle PRQ$ and $\angle PRT$ form a linear pair.
We know that sum of the angles of a linear pair is $180^{\circ}$
We know that sum of the angles of a linear pair is ${180^\circ }$
$\therefore \angle PQS + \angle PQR = {180^\circ },$ and $...(i)$
$\angle PRQ + \angle PRT = {180^\circ }....(ii)$
From equation $(i)$ and $(ii)$, we can conclude that
$\angle P Q S+\angle P Q R=\angle P R Q+\angle P R T$
$\text { But, } \angle PQR=\angle PRQ$
$\therefore \angle PQS=\angle PRT$
Hence, proved.
We are given that $\angle PQR =\angle PRQ$
From the given figure, we can conclude that $\angle PQS$ and $\angle PQR$, and $\angle PRQ$ and $\angle PRT$ form a linear pair.
We know that sum of the angles of a linear pair is $180^{\circ}$
We know that sum of the angles of a linear pair is ${180^\circ }$
$\therefore \angle PQS + \angle PQR = {180^\circ },$ and $...(i)$
$\angle PRQ + \angle PRT = {180^\circ }....(ii)$
From equation $(i)$ and $(ii)$, we can conclude that
$\angle P Q S+\angle P Q R=\angle P R Q+\angle P R T$
$\text { But, } \angle PQR=\angle PRQ$
$\therefore \angle PQS=\angle PRT$
Hence, proved.
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