Q. 214.1( 7 Votes )

In ∆ABC, AB=AC and the bisectors of B and C meet at a point O. prove that BO=CO and the ray AO is the bisector of A.


Answer :

Given: In ∆ABC, AB=AC and the bisectors of B and C meet at a point O.


To prove: BO=CO and BAO = CAO


Proof:


In , ∆ABC we have,


OBC = B


OCB = C


But B = C … given


So, OBC = OCB


Since the base angles are equal, sides are equal


OC = OB …(1)


Since OB and OC are bisectors of angles B and C respectively, we have


ABO = B


ACO = C


∴∠ABO = ACO …(2)


Now in ∆ABO and ∆ACO


AB = AC … given


ABO = ACO … from 2


BO = OC … from 1


Thus by SAS property of congruence,


∆ABO ∆ACO


Hence, we know that, corresponding parts of the congruent triangles are equal


BAO = CAO


ie. AO bisects A


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