Question

In triangle ABC, the bisectors of angle ABC and angle BCA intersect each other at O. Find the measure of angle BOC.​

Answer 1

Answer:

∴ ∠BOC = $\frac{1}{2}$ ∠A + 90°

Step-by-step explanation:

Given:

In triangle ABC, OB and OC are bisectors of ∠B  and  ∠C

 ∴     ∠B = 2 ∠OBC

and  ∠C = 2 ∠OCB

Since,  ∠A + ∠B + ∠C = 180°       (1)

Putting value of ∠B and ∠C  in equation (1), We get:

⇒ ∠A + 2 ∠OBC + 2 ∠OCB = 180°

⇒ ∠A + 2 ( ∠OBC + ∠OCB ) = 180°         (2)  

Now,

In triangle OBC,

⇒ ∠BOC + ∠OBC + ∠OCB = 180°

∴  ∠OBC + ∠OCB = 180°  − ∠BOC          (3)

Now, putting value of equation (2) in equation (3)

⇒ ∠A + 2 ( 180°   − ∠BOC ) = 180°  

⇒ ∠A + 360°  − 2 ∠BOC = 180°  

⇒ ∠A + 180°  = 2 ∠BOC

∴ ∠BOC = $\frac{1}{2}$ ∠A + 90°