Question

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

Answer 1

Given:

$\frac{PS}{SQ}=\frac{PT}{TR}$ and $\angle PST=\angle PRQ$

To Prove: $PQR$ is isosceles triangle.

Proof:

Given,

$\frac{PS}{SQ}=\frac{PT}{TR}$

By converse of BPT we get,

$ST\parallel QR$

$\therefore \angle PST=\angle PQR$ (Corresponding angles) $\dots \left(i\right)$

But, $\angle PST=\angle PRQ$ (Given) $\dots \left(ii\right)$

From equation (i) and (ii)

$\angle PQR=\angle PRQ\Rightarrow PR=PQ$

Hence$,\Delta PQR$ is an isosceles triangle.