Question

In given figure, $\frac{PS}{SQ}=\frac{PT}{TR}$ and $\angle PST=\angle PRQ$. Prove that $△PQR$ is an isosceles triangle.

Answer 1

We have,
$\frac{PS}{SQ}=\frac{PT}{TR}$

$\Rightarrow$ $ST||QR$ [By using the converse of Basic Proportionality Theorem]

$\Rightarrow$ $\angle PST=\angle PQR$ [Corresponding angles]

$\Rightarrow$ $\angle PRQ=\angle PQR$ [$\because \angle PST=\angle PRQ$ (Given)]

$\Rightarrow$ $PQ=PR$ [$\because$ Sides opposite to equal angles are equal]

$\Rightarrow$ $△PQR$ is isosceles.