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The bisectors of angleB and angleC of an...

The bisectors of `angleB and angleC` of an isosceles triangle with `AB = AC` intersect each other at a point `O. BO` is produced to meet `AC` at a point `M.` Prove that `angleMOC = angleABC.`

Text Solution

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Given Lines,OB and OC are the angle bisectors of `angleB and angleC ` of an isoceles `Delta ABC` such that AB=AC which intersect each other at O and BO is produced to M.
To Prove
`angleMOC=angleABC`

Proof In `Delta ABC " " AB=AC" "["given"]`
`rArr " "angleACB=angleABC`[angles opposite to equal sies are equal ]
`(1)/(2)angleACB=(1)/(2)angleABC " "["dividing both sides by 2 "]`
`rArr " "angleOCB=angleOBC `
[exterior angle of a triangle is equal to the sum of two interior of two interior angles ]
`rArr " "angleMOC=angleOBC+angleOBC " "["from Eq.(i)"]`
`rArr " "angleMOC=2angleOBC`
`rArr " "angleMOC=angleABC " "["since,OB is the bisector of" angleB]`
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