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Question
In ΔABC, AB = AC and the bisectors of angles B and C intersect at point O.Prove that BO = CO and the ray AO is the bisector of angle BAC.
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Solution
In ΔABC,
Since AB = AC
∠C = ∠B ...(angles opposite to the equal sides are equal)
BO and CO are angle bisectors of ∠B and ∠C respectively
Hence, ∠ABO = ∠OBC = ∠BCO = ∠ACO
Join AO to meet BC at D
In ΔABO and ΔACO and
AO = AO
AB = AC
∠C = ∠B =
Therefore, ΔBAO ≅ ΔACO ...(SAS criteria)
Hence, ∠BAO = ∠CAO
⇒ AO bisects angle BAC
In ΔABO and ΔACO
and AB = AC
AO = AO
∠BAD = ∠CAD = ...(proved)
ΔBAO ≅ ΔACO ...(SAS criteria)
Therefore,
BO = CO.
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