In triangle ABC, the degree measure of the angle BAC is 45 degrees, Find the degree measure of the Angle BOC

In triangle ABC, the degree measure of the angle BAC is 45 degrees, Find the degree measure of the Angle BOC, where O is the intersection point of the bisketrices BL and CM of triangle ABC

1. In triangle ABC we denote angle B as x, angle C as y. By the theorem on the sum of the angles of a triangle:
angle A + angle B + angle C = 180 degrees;
45 + x + y = 180;
x + y = 180 – 45;
x + y = 135.
2. Since BL is the bisector of angle B, then:
angle ABL = angle LBC (aka angle OBC) = angle B / 2 = x / 2.
Since CM is the bisector of angle C, then:
angle ВСМ (aka angle ВСО) = angle МСА = angle С / 2 = у / 2.
3. Consider a triangle BOC: angle OBC = x / 2, angle BCO = y / 2. By the theorem on the sum of the angles of a triangle:
OBC angle + BCO angle + BOC angle = 180 degrees;
x / 2 + y / 2 + angle BOC = 180;
angle BOС = 180 – (x + y) / 2.
Since x + y = 135, then:
angle ВOС = 180 – 135/2;
angle ВOС = (360 – 135) / 2;
angle ВOС = 225/2;
angle ВOС = 112.5 degrees.
Answer: ВOС angle = 112.5 degrees.