# OA = OB, OC = OD, Angle AOB = Angle COD. Prove that AC=BD

Dear Student,

Given : $\angle$ AOB =  $\angle$ COD                              --- ( 1 )

And

$\angle$ COB  =  $\angle$ COB                                          --- ( 2 )     (  Same angles )

So, If we subtract equation 2 in equation 1 we get  :

$\angle$ AOB - $\angle$ COB =  $\angle$ COD - $\angle$ COB

We know " Euclid's third Axiom: If equals be subtracted from equals, the remainders are equal. "  we get :

$\angle$ AOC  =  $\angle$ BOD                                          --- ( 3 )

In $∆$ AOC and $∆$ BOD

OA  =  OB                                                      ( Given )

$\angle$ AOC  =  $\angle$ BOD                                      ( From equation 3 )

And

OC  =  OD                                                      ( Given )

Thus ,

$∆$ AOC $\cong$ $∆$ BOD                                        ( By SAS rule )

Therefore,

AC  =  BD                                                      ( By CPCT )                           ( Hence proved )

If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible.

Regards

• 2
Hi,

Given :  ∠  AOB  =  ∠  COD  ,
Now we subtract  ∠  COB from both hand side we get

∠  AOB  -   ∠  COB  =   ∠  COD  -   ∠  COB  ,

∠  AOC  =   ∠  BOD   -- ( 1 )

In  ∆  AOC and  ∆  BOD

OA  =  OB   ( Given )

∠  AOC  =   ∠  BOD    ( From equation 1 )

And

OC  =  OD    ( Given )

So,

∆  AOC  ≅   ∆  BOD  ( By SAS rule )

Therefore,

AC  =  BD    ( By CPCT) …   ( Hence proved )

Cheers!!
• 1
We havent learnt the subtraction part and stuff. So i probably didnt understand. Well thanks for ur help. But can someone do it in an easier way?
• 0
Hi,

@ Arwa: The subtraction part to get the angles of two triangles, is based on understanding. So, as Angle COB was common in triangle AOB and COD, so we subtracted that.  Also, such questions can only be solved by congruency rules. Thus, this is the standard method to solve.

Important tip (as per the chapter): Learn and revise the congruence rules which needs to be applied in different cases. Solve all the examples and questions, given in the N.C.E.R.T. .

Best Wishes!!
• 1
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