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  • In Fig. 6.9, OD is the bisector of , OE is the bisector of and  Show that the points A, O and B are collinear.

In Fig. 6.9, 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.

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Solution.

Given in the figure OD \perp OE., OD and OE are bisectors of \angle AOC and \angle BOC respectively
To show: Points A, O & B are collinear, i.e., AOB is a straight line
Proof: Since OD and OE bisect angles \angle AOC and \angle BOC respectively
\angle AOC = 2 \angle DOC… (i)
And \angle COB = 2 \angle COE … (ii)
On adding equation (i) and (ii) we get
\angle AOC + \angle COB = 2 \angle DOC + 2 \angle COE
\angle AOC + \angle COB = 2 (\angle DOC + \angle COE)
\angle AOC + \angle COB = 2 \angle DOE
\angle AOC + \angle COB = 2 \times 90^{\circ} (Given OD \perp OE)
\angle AOC + \angle COB = 180^{\circ}
\angle AOB = 180^{\circ}
So, \angle AOC & \angle COB are forming a linear pair.
AOB is a straight line.
Therefore, points A, O and B are collinear.

Hence proved

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