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Question
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.
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Solution
Given in the question, bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Now BO is produced to a point M.
In triangle ABC,
AB = AC
∠ABC = ∠ACB ...[Angle opposite to equal sides of a triangle are equal]
`1/2 ∠ABC = 1/2 ∠ACB`
That is ∠1 = ∠2 ...[Since, BO and CO are bisectors of ∠B and ∠C]
In triangle OBC,
Exterior ∠MOC = ∠1 + ∠2 ...[Exterior angle of a triangle is equal to the sum of interior opposite angles]
Exterior ∠MOC = 2∠1 ...[∠1 = ∠2]
Hence, ∠MOC = ∠ABC.
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