Question

if one of the angle in a triangle is 120 degree then the angle between the bisectors of other two angles can be ​

Answer 1

Sum of all angles of a triangle is 180°.

If angle A = 120° , then angles B+C = 60

In the triangle formed by the intersection of the bisectors of angles B and C the sum of interior angles is 180°.

therefore Angles O + (B/2) + (C/2) = 180°

=> O + {(B+C)/2} = 180

=> O + 30 = 180

=> O = 150°

Answer 2

Given,

One angle of a triangle is 120°.

To find,

The angle between the bisectors of the other two angles.

Solution,

We can solve this problem simply by following the process given below.

First of all, let there be a ΔABC, for which ∠A (or ∠BAC) is 120°, and let the bisectors OB and OC of ∠B (or ∠ABC) and ∠C (or ∠ACB) respectively intersect at O. So, we have to find this ∠BOC.

Further, suppose that ∠B = 2x, ∠C = 2y, and ∠BOC = z.

Now, consider ΔABC, as the sum of all angles of a triangle is 180°, so,

∠ABC + ∠ACB + ∠ABC = 180, or,

∠B + ∠C + ∠A = 180

$2x+2y+120=180$ (since, ∠A = 120)

⇒ $2x+2y=60$

⇒ $x+y=30$

Now consider the ΔBOC, formed by bisectors OB, OC, and side BC of ΔABC. Here, the sum of ∠OBC, ∠OCB, and ∠BOC will be 180°.

Also, as ∠ABC = 2x and ∠ACB = 2y, and OB, OC are their respective bisectors, so ∠OBC = x and ∠OCB = y. Thus,

$x+y+z=180$

Substituting the value for x+y that is 30, from the previous expression, in the above equation, we get,

$30+z=180$

⇒ $z=180-30$

⇒ $z=150$.

Or, ∠BOC = 150°.

Therefore, the angle between bisectors of the other two angles of the given triangle will be 150°.