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Question

The bisectors of B and C of an isosceles triangle with AB = AC intersect each other at a point O. BO is produced to meet AC at a point M Prove that MOC = ABC.

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Solution

Given: In isosceles ABC,AB=AC;OBandOCarebisectorsofBandC, respectively.
To prove: MOC=ABC
Proof:
In ABC ,
∵ AB = AC
(Given)
ABC=ACB (Angles opposite to equal sides are equal)
⇒12ABC=12ACB
OBC=OCB
(Given, OB and OC are the bisectors of BandC, respectively) .....(i)
Now, in OBC,MOC is an exterior angle
MOC=OBC+OCB (An exterior angle is equal to the sum of two opposite interior angles)
MOC=OBC+OBC [From (i)]
MOC=2OBC
Hence, MOC=ABC(Given,OBisthebisectorofB)


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