Question

In ΔPQR, ST is a line such that PS/SQ = PT/TR and also ∠PST = ∠PRQ. Prove that ΔPQR is an isosceles triangle.

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

PS = PT.....i
SQ = TR.........ii

Adding i and ii...........

PS+SQ = PT+TR
PQ = PR

Hence, triangle PQR is isosceles........