Question

OD is the bisector of angle AOC ,OE is the bisector of angle BOC and OD is perpendicular to OE .show that AOB is collinear

Answer 1

Hope it helps!!!!!

Let ∠ AOC  =  2x 

So,

∠ AOD =  ∠ DOC  = x    
( As given OD is the bisector of ∠ AOC ) 

Let ∠ BOC  =  2y 

So,

∠ BOE  =  ∠ EOC  =  y                                               (  As given OE is bisector of ∠ BOC ) 

And,

∠ DOE  = 90°                                                              ( As given OD is perpendicular to OE  ) 

We can write ∠ DOE ,

As  : 

∠ DOE  =  ∠ DOC  + ∠ EOC  

∠ DOC  +  ∠ EOC  = 90​°               ------- ( 1 )                   ( As given ∠ DOE  = 90°   ) 

x  + y  =  90​°                                                               ( From our assumption )

∠ AOD  +  ∠ BOE  =  x  +  y  

So,

∠ AOD +  ∠ BOE  = ​ 90​°            --------- ( 2 )

Now we add equation 1 and 2 we get 

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

So we can say that A , O and B are colinear .                         ( Hence proved )

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Answer 2

Show that AOB is collinear

Step-by-step explanation:

Given that,

OE is the bisector of ∠BOC

OD is the bisector of ∠AOC

while OD ⊥ OE,

To prove:

AOB is collinear

Proof:

∠AOD = ∠COD (OD is the bisector of ∠AOD)

∠BOE = ∠COE (OE is the bisector of ∠BOC)

SO, ∠AOC = 2∠DOC

∠BOC = 2∠EOC

∠AOC + ∠BOC = 2(∠DOC + ∠EOC)

= 2 * 90°

= 180°

∵ AOB is a straight line and thus, AOB is collinear.

Learn more: Collinear points

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