Question

In $\Delta PQR$, the angle bisectors of $\angle PQR$ and $\angle PRQ$ meet at O.

If $\angle QPR={80}^{\circ }$, find the measure of $\angle QOR$

• A. ${80}^{\circ }$
• B. ${130}^{\circ }$
• C. ${100}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is B ${130}^{\circ }$


$\angle 1=\angle 2=\frac{1}{2}\angle Q$ ($OQ$ bisects $\angle PQR$)
$\Rightarrow \angle Q=2\angle 2$
$\angle 3=\angle 4=\frac{1}{2}\angle R$ ($OR$ bisects $\angle PRQ$)
$\Rightarrow \angle R=2\angle 3$
In $\Delta PQR,{80}^{\circ }+\angle Q+\angle R={180}^{\circ }$
$\Rightarrow \angle Q+\angle R={100}^{\circ }$
$\Rightarrow 2\angle 2+2\angle 3={100}^{\circ }$
$\Rightarrow 2(\angle 2+\angle 3)={100}^{\circ }$
$\Rightarrow \angle 2+\angle 3={50}^{\circ }$
In $\Delta QOR,\angle 2+\angle 3+\angle QOR={180}^{\circ }\Rightarrow {50}^{\circ }+\angle QOR={180}^{\circ }$
$\Rightarrow \angle QOR={180}^{\circ }-{50}^{\circ }={130}^{\circ }$