Question

In triangle ABC, the bisectors of angle B and C meet at O. Find the value of angle BOC in terms of angle A​

Answer 1

Answer:

BOC = 90 + 1/2*(BAC)

Step-by-step explanation:

Let us assume that bisector for B divides angle ABC into 2 equals angles, let say x and  bisector for ACB divides angle  into 2 equals angles, let say y.

Therefore angle ABC = 2x and angle ACB = 2y

Now consider Triangle BOC,

since sum of angles of triangle is equal to 180, thus,

BOC + OBC +OCB = 180

=>x + y + BOC = 180

=> x + y = 180 - BOC -----------(i)

Similarly in triangle ABC

ABC + ACB + BAC = 180

=> 2x + 2y + BAC = 180

=> 2(x+ y) + BAC = 180

=> 2(180 - BOC) + BAC =  180             (from i)

=> 360 - 2BOC + BAC = 180

Rearranging above equation we get,

180 + BAC = 2BOC

Now dividing each term by 2, we get,

BOC = 90 + 1/2*BAC   (answer)