Question

In fig.OD bisects Angle AOC and OC bisects Angle BOC .Prove that A,O,B are collinear.

Answer 1

Answer:

In the attached figure,

∵ OD bisects ∠AOC

∴ ∠AOD = ∠DOC

Similarly ∵ OE bisects ∠BOC

∴ ∠BOE = ∠EOC

Given that OD⊥OE

∴ ∠DOC + ∠EOC = 90°

∴∠AOD + ∠BOE = 90°    

∴ ∠AOD + ∠DOC + ∠BOE + ∠EOC = 90° + 90° = 180°

∴ by linear pair axiom, OD and OE are rays on line AB

Therefore, points A, O, B are collinear

Answer 2

Answer:

Step-by-step explanation:

Our question is: In fig.OD bisects Angle AOC and OC bisects Angle BOC.

Our aim is to prove that A,O,B are collinear.

In order to answer this question we need to define collinear point. What are collinear points? Collinear points are those who lay in the same line.

Hence, what we have in the attached figure is that:

• OD is a bisector of ∠AOC.

• ∠AOD = ∠DOC.

• Also, OE is a bisector of ∠BOC.

Hence we conclude that ∠BOE = ∠EOC.

We are also given, from the information above and from the picture, that OD⊥OE. Thus:

• ∠DOC + ∠EOC = 90°

• ∠AOD + ∠BOE = 90°    

• ∠AOD + ∠DOC + ∠BOE + ∠EOC = 90° + 90° = 180°

Hence by linear pair axiom, we can conclude that OD and OE are rays on line AB.

Hence points A, O, B are collinear

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