Question

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

$\text{If}\angle QPR={80}^{\circ },\text{find the measure of}\angle QOR$.
[3 Marks]

Answer 1

$\angle 1=\angle 2=\frac{1}{2}\angle Q\text{(OQ bisects}\angle PQR$)

$\Rightarrow \angle Q=2\angle 2$

$\angle 3=\angle 4=\frac{1}{2}\angle R\text{(OR bisects}\angle PRQ$)

$\Rightarrow \angle R=2\angle 3$ [1 Mark]

$\text{In}\Delta PQR,{80}^{\circ }+\angle Q+\angle R={180}^{\circ }$
$\Rightarrow \angle Q+\angle R={100}^{\circ }$
$\Rightarrow 2\angle 2+2\angle 3={100}^{\circ }$
$\Rightarrow 2(\angle 2+\angle 3)={100}^{\circ }$
$\Rightarrow \angle 2+\angle 3={50}^{\circ }$ [1 Mark]

$\text{In}\Delta QOR,\angle 2+\angle 3+\angle QOR={180}^{\circ }\Rightarrow {50}^{\circ }+\angle QOR={180}^{\circ }$
$\Rightarrow \angle QOR={180}^{\circ }-{50}^{\circ }={130}^{\circ }$ [1 Mark]