In Fig. 6.44, the

To Prove: ∠QTR = 1/2 ∠QPR

Given: Bisectors of ∠PQR and ∠PRS meet at point T

Proof:

In ΔQTR,

∠TRS = ∠TQR + ∠QTR (Exterior angle of a triangle equals to the sum of the two opposite interior angles)

∠QTR = ∠TRS - ∠TQR -----------(i)

Similarly in ΔQPR,

∠SRP = ∠QPR + ∠PQR                        

2∠TRS = ∠QPR + 2∠TQR  (∵ ∠TRS and ∠TQR are the bisectors of ∠SRP and ∠PQR respectively.) 

∠QPR = 2 ∠TRS - 2 ∠TQR

 .............(ii)

From (i) and (ii), we get

Hence, proved