(Q17) In ∆PQR, ST is a line such that

PS

SQ

=

PT

TR

and also ∠PST = ∠PRQ.

Prove that ∆PQR is an isosceles triangle.

Solution :

Given : In ∆PQR,

PS

SQ

=

PT

TR

and ∠PST = ∠PRQ.

R.T.P : ∆PQR is Isosceles.

Proof :

PS

SQ

=

PT

TR

Hence, ST ∥ QR (Converse of Basic proportionality theorem)

∠PST = ∠PRQ .....(1)

(Corresponding angles for the lines ST ∥ QR)

Also, ∠ = ∠PRQ .....(2) given

From (1) and (2),

∠PQR = ∠

i.e., =

[∵ In a triangle sides opposite to equal angles are equal]

Hence, ∆PQR is an isosceles triangle.