In ∆PQR, ST is a line such that \frac{PS}{SQ} = \frac{PT}{TR} and also \angle PST = \angle PRQ . Prove that ∆PQR is an isosceles triangle

Step 1: Understand the question and note down the given data
NOTE: In triangle PQR, \frac{PS}{SQ} = \frac{PT}{TR}
\angle PST = \angle PRQ
Step 2: Remember the converse of the basic the proportionality theorem
NOTE: If a line divides two sides of a triangle in the same ratio, then the
line is parallel to the third side.
EXAMPLE: In triangle PQR, \frac{PS}{SQ} = \frac{PT}{TR}
So, ST II QR
Step 3: Find the angles of the triangles
EXAMPLE: ST II QR, QP and PR are tranversals
Therefore, ∠ PST = ∠ PQR (Corresponding angles)
Also it is given that ∠ PST = ∠ PRQ
So, ∠ PRQ = ∠ PQR
Step 4: Prove that the Δ PQR is an isosceles triangle.
∠ PRQ = ∠ PQR (prove in step 3)
Therefore, PQ = PR ( sides opposite the equal angles)
So, Δ PQR is an isosceles triangle.
Hence proved.