$\frac{P S}{S Q}$ = $\frac{P T}{T R}$ and $∠$ PST = $∠$ PRQ. Prove that $Δ$ PQR is an isosceles triangle.

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Solution

It is given that $\frac{P S}{S Q}$ = $\frac{P T}{T R}$ ⋅

So, ST || QR (Theorem)

$∴$ $∠$ PST = $∠$ PQR (Corresponding angles) (1)

Also, it is given that

$∠$ PST = $∠$ PRQ (2)

So, $∠$ PRQ = $∠$ PQR [From (1) and (2)]

$∴$ PQ = PR (Sides opposite the equal angles)

i.e., PQR is an isosceles triangle.

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