In a δAbc, It is Given that Ab = Ac and the Bisectors of ∠B and ∠C Intersect at O. If M Is A Point on Bo Produced, Prove that ∠Moc = ∠Abc. - Mathematics

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In a ΔABC, it is given that AB = AC and the bisectors of ∠B and ∠C intersect at O. If M is a point on BO produced, prove that ∠MOC = ∠ABC. 

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Given that in , ΔABC, 

AB=AC and the bisector of ∠B and ∠C intersect at O and M is
a point on BO produced  

We have to prove ∠ MOC=∠ABC 


AB =AC ⇒ΔABC is isosceles ⇒ ∠B=∠C(or) ∠ABC=∠ACB 


BO and CO are bisectors of ∠ABC and ∠ACB  respectively

 ⇒ABO=∠OBC=∠ACO=∠OB=`1/2` ∠ABC=`1/2`∠ACB ............(1) 

We have, in ΔOBC 

∠OBC +∠OCB +∠BOC =180°          .............(2) 

And also 

∠BOC +∠COM =180°             ..................(3)[Straight angle] 

Equating (2) and (3) 


⇒ ∠OBC+∠OBC=∠MOC           [∵ from (1)] 

⇒ 2∠OBBC=∠MOC 

⇒2`(1/2∠ABC)`=∠MOC           [∵from (1)]  

⇒ ∠ABC=∠MOC 


Concept: Properties of a Triangle
  Is there an error in this question or solution?


RD Sharma Mathematics for Class 9
Chapter 12 Congruent Triangles
Exercise 12.3 | Q 8 | Page 47

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