Question

in the adjoining figure ,OD is bisector of AOC ,OE is bisector of BOC and OD perpendicular to OE . show that the points A,O and B are collinear

Answer 1

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Given In the figure, OD ⊥ OE, OD and OE are the bisectors of ∠AOC and ∠BOC.
To show Points A, O and B are collinear i.e., AOB is a straight line.
Proof Since, OD and OE bisect angles ∠AOC and ∠BOC, respectively.
∠AOC =2 ∠DOC …(i)
and ∠COB = 2 ∠COE …(ii)
On adding Eqs. (i) and (ii), we get
∠AOC + ∠COB = 2 ∠DOC +2 ∠COE ⇒ ∠AOC +∠COB = 2(∠DOC +∠COE)
⇒ ∠AOC + ∠COB= 2 ∠DOE
⇒ ∠AOC+ ∠COB = 2 x 90° [∴ OD ⊥ OE]
⇒ ∠AOC + ∠COB = 180°
∴ ∠AOB = 180°
So, ∠AOC and ∠COB are forming linear pair.
Also, AOB is a straight line.
Hence, points A, O and B are collinear.

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Answer 2

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