Question

In figure $OE$ bisects $\angle AOC,OF$ bisects $\angle COB$ and $OE\perp OF$. Show that $A,O,B$ are collinear or $AOB$ is a straight line.

Answer 1

Since $OE$ bisects $\angle AOC$

So, $\angle AOE=\angle EOC$

$\angle EOC=\frac{1}{2}\angle AOC$ --- (i)

also ,$OF$ bisects $\angle COB$

$\angle COF=\angle BOF$

$\angle COF=\frac{1}{2}\angle BOC$ ----- (ii)

Now, $OE\perp OF$

$\angle EOF={90}^{\circ }$

$\angle EOC+\angle COF={90}^{\circ }$

$\frac{1}{2}\angle AOC+\frac{1}{2}\angle BOC={90}^{\circ }$ from (i) and (ii)

$\angle AOC+\angle BOC={180}^{\circ }$

$AOB$ is straight line

Prove.