Question

In ∆ABC, the bisectors of ∠ABC and ∠BCA intersect each other at O. The measure of ∠BOC is:

Answer 1

In the given figure, the bisectors of ABC and BCA intersect each other at the point O . Prove that BOC = 90 + 12 A

Answer 2

$\sf\huge\bold{Answer:}$

Given :

∆ABC

the bisectors of ∠ABC and ∠BCA intersect each other at O.

To find :

Measure of ∠BOC

Solution :

In ∆ ABC,

:⟶∠A+∠B+∠C=180°...(i)

where,

OB and OC are bisectors of ∠B  and  ∠C

 ⇒∠B=2 ∠OBC

⇒∠C=2 ∠OCB

Now equation (i) can be written as,

∠A+2(∠OBC+∠OCB)=180° ...  (ii)

In ∆ OBC,

∠BOC+∠OBC+∠OCB=180°

∠OBC+∠OCB=180°−∠BOC...(iii)

From (ii)  and (iii),

∠A+2(180°−∠BOC)=180°

∠A+360°−2∠BOC=180°

∠A+180°=2∠BOC

½∠A+90°=∠BOC

$\purple{∠BOC=½∠A+90°}$