Question

In a fig, the side QR of $\Delta PQR$ is produced to a point S. If the bisectors of $\angle PQR$ and $\angle PRS$ meet at point T, then prove that $\angle QRT=\frac{1}{2}\angle QPR$.

Answer 1

Given: In a fig, the side QR of $\Delta PQR$ is produced to a point S.

The bisectors of $\angle PQR$ and $\angle PRS$ meet at point T.

To Prove: $\angle QTR=\frac{1}{2}\angle QPR$.

Proof:

$\angle PRS=\angle PQR+\angle QPR$ ....(1)

(Sum of interior opposite angles is equal to the exterior angle).

$\angle TRS=\angle TQR+\angle QTR$ ....(2)

$\Rightarrow 2\angle TRS=2\angle TQR+2\angle QTR$

$\Rightarrow \angle PRS=\angle PQR+2\angle QTR$

(OT bisects $\angle PQR$ and RT bisects $\angle PRS$)

$\angle PQR+\angle QPR=\angle PQR+2\angle QTR$

(From 1 and 2)

$\Rightarrow \angle QPR=2\angle QTR$

or $\angle QTR=\frac{1}{2}\angle QPR$ $\left[henceproved\right]$