Question

In 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

$\angle QTR=\frac{1}{2}\angle QPR$ statement is

• A. True
• B. False

Answer 1

The correct option is A True

It is given that bisectors of $\angle PQR$ and $\angle PRS$ meet at point $T.$

$\Rightarrow$ $\angle PRT=\angle SRT$ [ Since, $T$ is an angle bisector of a $\angle PRS$ ]

$\Rightarrow$ $\angle PQT=\angle TQR$ [ Since, $T$ is an angle bisector of a $\angle PQR$ ]

We know, the exterior angle of a triangle equals the sum of the two interior angles.

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

$\Rightarrow$ $\angle QTR=\angle TRS-\angle TQR$ ---- ( 1 )

An exterior angle of a triangle equals the sum of the two interior angles.

$\Rightarrow$ $\angle SRP=\angle QPR+\angle PQR$

$\Rightarrow$ $2\angle TRS=\angle QPR+2\angle TQR$ [ $TR$ is a bisector of $\angle SRP$ and $QT$ is a bisector of $\angle PQR$ ]

$\Rightarrow$ $\angle QPR=2\angle TRS-2\angle TQR$

$\Rightarrow$ $\angle QPR=2(\angle TRS-\angle TQR)$

$\Rightarrow$ $\frac{1}{2}\angle QPR=\angle TRS-\angle TQR$ ----- ( 2 )

Equating ( 1 ) and ( 2 ) we get,

$\Rightarrow$ $\angle QTR=\frac{1}{2}\angle QPR$