Question

In the figure, the side QR of ∆PQR is produced to a point S. If the bisectors of ∠ PQR and ∠ PRS meet at point T, then prove that ∠QTR = 1/ 2 ∠QPR.

Answer 1

Exterior ∠PRS = ∠PQR + ∠QPR 

[Exterior angle property] 

Therefore, 1 2 ∠PRS = 1 2 ∠PQR + 1 2 ∠QPR 

⇒ ∠TRS = ∠TQR + 1 2 ∠QPR …(i) 

But in ∆QTR, 

Exterior ∠TRS = ∠TQR + ∠QTR …(ii) 

[Exterior angles property] Therefore, from (i) and (ii) 

∠TQR + ∠QTR = ∠TQR + 1 2 ∠QPR 

⇒ ∠QTR = 1 2 ∠QPR Proved.