In the figure,$\frac{PS}{SQ} = \frac{PT}{TR}$ and $\angle PST = \angle PRQ$. Prove that $\triangle PQR$ is an isosceles triangle.
(N/A) It is given that $\frac{PS}{SQ} = \frac{PT}{TR}$.
By the Converse of Thales Theorem (Basic Proportionality Theorem),$ST \parallel QR$.
Therefore,$\angle PST = \angle PQR$ (Corresponding angles) $...(1)$
Also,it is given that $\angle PST = \angle PRQ$ $...(2)$
From equations $(1)$ and $(2)$,we get $\angle PQR = \angle PRQ$.
Since the angles opposite to the sides $PQ$ and $PR$ are equal,the sides themselves must be equal.
Therefore,$PQ = PR$.
Hence,$\triangle PQR$ is an isosceles triangle.
By the Converse of Thales Theorem (Basic Proportionality Theorem),$ST \parallel QR$.
Therefore,$\angle PST = \angle PQR$ (Corresponding angles) $...(1)$
Also,it is given that $\angle PST = \angle PRQ$ $...(2)$
From equations $(1)$ and $(2)$,we get $\angle PQR = \angle PRQ$.
Since the angles opposite to the sides $PQ$ and $PR$ are equal,the sides themselves must be equal.
Therefore,$PQ = PR$.
Hence,$\triangle PQR$ is an isosceles triangle.
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