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In the given figure, $\frac{PS}{SQ} = \frac{PT}{TR}$ and $\angle PST = \angle PRQ$. Prove that $\triangle PQR$ is an isosceles triangle.
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Given $\frac{PS}{SQ} = \frac{PT}{TR}$
$\Rightarrow ST || QR$ ($1$)
$\therefore \angle PST = \angle PQR$ ($1/2$)
and given, $\angle PST = \angle PRQ$
So, $\angle PQR = \angle PRQ$
$\therefore \triangle PQR$ is an isosceles triangle. ($1/2$)
$\Rightarrow ST || QR$ ($1$)
$\therefore \angle PST = \angle PQR$ ($1/2$)
and given, $\angle PST = \angle PRQ$
So, $\angle PQR = \angle PRQ$
$\therefore \triangle PQR$ is an isosceles triangle. ($1/2$)