Question

Question 1
In the figure, OD is the bisector of $\angle AOC$, OE is the bisector of $\angle BOC$ and OD $\perp$ OE. Show that the points A, O and B are collinear.

Answer 1

Given, OD $\perp$ OE, OD and OE are the bisectors of $\angle AOCand\angle BOC$.
To show points A, O and B are collinear, i.e., AOB is a straight line.
Since OD and OE bisect angles $\angle AOCand\angle BOC$ respectively,. . . . . . . .(i)
$\therefore \angle AOC=2\angle DOC$.......(i)
$and\angle COB=2\angle COE$..........(ii)
On adding equations (i) and (ii), we get,
$\angle AOC+\angle COB=2\angle DOC+2\angle COE$
$\Rightarrow \angle AOC+\angle COB=2(\angle DOC+\angle COE)$
$\Rightarrow \angle AOC+\angle COB=2\angle DOE$
$\Rightarrow \angle AOC+\angle COB=2\times {90}^{\circ }$
$\Rightarrow \angle AOC+\angle COB={180}^{\circ }$
$\therefore \angle AOB={180}^{\circ }$
$\angle AOCand\angle COB$ are forming linear pair.
Therefore, AOB is a straight line.
Hence, points A, O and B are collinear.