Question

$\text{The bisectors of}\angle Band\angle C\text{of an isosceles triangle with AB = AC intersect each other at a point O. BO is produced to meet AC at a point M Prove that}\angle MOC=\angle ABC.$

Answer 1

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