The size of the acute angle formed by the bisectors of angles B and C in triangle ABC is 30 degrees, as derived from using the properties of angle bisectors within a triangle.
To find the size of the acute angle formed by the bisectors of angle B and angle C in triangle ABC, we must first determine the measure of each bisected angle. Since angle B is 40 degrees and angle C is 60 degrees, their bisectors will divide these angles equally, creating two angles of 20 degrees and 30 degrees respectively.
From the properties of triangle ABC, we know that the sum of angles in a triangle is 180 degrees. Therefore, if we subtract the sum of angles B and C from 180 degrees, we get angle A: 180 - (40 + 60) = 80 degrees.
When the angle bisectors are drawn within the triangle, they meet at the incenter, which is equidistant from all sides of the triangle. The angle at the incenter opposite angle A is the exterior angle to the acute angle we are trying to find. Following the exterior angle property, the angle at the incenter is the sum of the opposite interior angles, which means it is 20 + 30 = 50 degrees.
The acute angle formed by the two angle bisectors within the triangle will then be 80 - 50 = 30 degrees, since the exterior angle at the incenter and the angle formed by the bisectors at the incenter are supplementary.