In triangleABC,AB is equal to AC ,and the bisector of angle B and C intersect at point O prove that BO=CO and the ray AO is the bisector of angleBAC

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Solution

Given AB=AC
so angle ABC= angle ACB
As BO and CO are angular bisectors, we can say that angle OBC= angle OCB
so BOC is an isoscles triangle
so BO=CO
In triangle ABD and ABC
AB=AC(given)
angle BAD = angle CAD(since AD passes through point O which is the point of intersection of angular bisectors)
AD=AD(since it is a common side)
so BD=DC. since corresponding sides will be equal
Now in triangle BOD and triangle COD
BO=CO(proven above)
OD=OD(common side)
BD=DC(proven above)
so triangle BOD is congruent to triangle COD
so corresponding angles will be equal
so Angle BOD= Angle COD
so OD is the angular bisector
so ray AO is the angular bisector of BOC

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