Question

In figure, ray $OE$ bisects $\angle AOC$ and $OF$ bisects $\angle COB$ and $OE\perp OF$. Then, state whether $A,O,B$ are collinear.

• A. True
• B. False

Answer 1

The correct option is A True

$OE$ bisects $\angle AOC$

$\Rightarrow \angle AOE=\angle EOC=\angle COE=\frac{1}{2}COA$ ........(1)

$OF$ bisects $\angle BOC$

$\angle COF=\angle FOB=\angle COF=\frac{1}{2}\angle BOC$ ...........(2)

$\angle EOF=\angle COE+\angle COF=90$ ....($OE$ is perpendicular to $OF$).........(3)

Subsituting (1) and (2) in (3), we get

$\frac{1}{2}\angle COA+\frac{1}{2}\angle BOC=90$

$\Rightarrow \frac{1}{2}[\angle COA+\angle COB]=90$

$\angle AOC+\angle COB=180$

$\angle AOC$ and $\angle COB$ form a linear pair.

Hence, $A,O,B$ are collinear.