Question

In triangle pqr st is a line such that ps/sq = pt/tr. angle pst = angle prq

Answer 1

Solution:-
It is given that PS/SQ = PT/TR
So, ST II QR (According to B.P.T)
Therefore, ∠ PST = ∠ PQR (Corresponding angles)
Also it is given that ∠ PST  = ∠ PRQ
So, ∠ PRQ = ∠ PQR
Therefore, PQ = PR ( sides opposite the equal angles)
So, Δ PQR is an isosceles triangle. 
Hence proved.

Answer 2

as by the given clues ,, we have;

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

angle PST = angle PRQ

ST || PR

angle PST = angle PQR ( corresponding angles )

angle PRQ= angle PQR

by triangle theroem the two line line opposite to the equal angles are equal..

PQ = PR

hence it is a isosceles triangle....