In this chapter we will derive formula to calculate the value of angle formed by joining two angle bisector of triangle.
Consider above triangle ABC in which BO & CO are angle bisectors dividing ∠B & ∠C into two equal parts.
These angle bisectors meet at point O.
The angle formed by angle bisector is given as;
∠BOC = 90 + 1/2 ∠A
Proof of angle bisector formula
In above triangle ABC, OB and OC are angle bisectors.
∠ 1 = ∠2
∠ 3 = ∠4
Applying angle sum property in triangle ABC.
∠A + ∠B + ∠C = 180
∠A + 2( ∠ 1) + 2( ∠3) = 180
2( ∠ 1) + 2( ∠3) = 180 – ∠A
∠1 + ∠3 = 90 – 1/2 ∠A
Now apply angle sum property in triangle OBC.
∠1 + ∠3 + ∠BOC = 180
90 – 1/2 ∠A + ∠BOC = 180
∠BOC = 90 + 1/2 (∠A)
Hence Proved.
Try to memorize the formula, as it would help you solve question fast.