Question

In the given figure the bisectors of angle ABC and angle BCA , intersect each other at point O . If angle BOC = 100° , the angle A is
(a) 30°
(b) 20°
(c) 40°
(d) 50°

Answer 1

WE KNOW THAT
bisector angles are equal

angles of triangle are twice the bisector angle

Answer 2

Given:

Bisectors of angle ABC and angle BCA , intersect each other at point O .

∠BOC = 100°

To find:

Measure of ∠A.

Solution:

Let the measure of ∠OBC be x and ∠OCB be y.

In triangle OBC, using angle sum property, we get

$\angle OBC+\angle OCB+\angle BOC=180^{\circ}$

$x+y+100^{\circ}=180^{\circ}$

$x+y=180^{\circ}-100^{\circ}$

$x+y=80^{\circ}$         ...(i)

Bisectors of angle ABC and angle BCA , intersect each other at point O .

So, ∠ABC is 2x and ∠ACB be 2y.

In triangle ABC, using angle sum property, we get

$\angle ABC+\angle ACB+\angle BAC=180$

$2x+2y+\angle A=180^{\circ}$

$2(x+y)+\angle A=180^{\circ}$

$2(80^{\circ})+\angle A=180^{\circ}$        [Using (i)]

$160^{\circ}+\angle A=180^{\circ}$

$\angle A=180^{\circ}-160^{\circ}$

$\angle A=20^{\circ}$

Therefore, the correct option is (b).