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


Given: Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O.


In ΔABC,


AB = AC (given)


ACB = ABC (opposite angles to equal sides are equal)


1/2 ACB = 1/2 ABC (divide both sides by 2)


OCB = OBC …(1) (As OB and OC are bisector of B and C)


Now, MOC = OBC + OCB (as exterior angle is equal to sum of two opposite interior angle)


⇒∠MOC = OBC + OBC (from (1))


MOC = 2OBC


⇒∠MOC = ABC (because OB is bisector of B)


Hence proved.


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