Triangle Angle Bisector Problem Explained

In geometry, when the internal angle bisectors of two angles in a triangle intersect, they meet at a special point called the incenter. This point is crucial for understanding various properties of triangles, especially those related to inscribed circles. The problem asks us to find the measure of the angle formed at this incenter when given one of the triangle's angles.

Angle Bisectors and Incenter Explained

• An angle bisector is a line segment that divides an angle into two equal smaller angles. For instance, if BI bisects ∠B, it means that ∠ABI and ∠CBI are equal. Similarly, CI bisects ∠C, meaning ∠BCI and ∠ACI are equal.
• The incenter (denoted as I in this problem) is the point where all three internal angle bisectors of a triangle intersect. It is unique because it is equidistant from all three sides of the triangle, serving as the center of the triangle's inscribed circle.
• The angle formed at the incenter by the intersection of two angle bisectors has a direct relationship with the third angle of the triangle.

Incenter Angle Formula Demystified

For any ΔABC, if the angle bisectors of ∠B and ∠C meet at point I (the incenter), the measure of ∠BIC can be determined using a specific formula. This formula connects the angle at the incenter to the angle opposite to the side that forms the base of the incenter angle (∠A in this case).

The formula is:

$$\angle\text{BIC} = 90^\circ + \frac{\angle\text{A}}{2}$$

This formula arises from applying the angle sum property of triangles. In ΔBIC, the sum of its angles is 180°. Since BI and CI are angle bisectors, ∠IBC = ∠B/2 and ∠ICB = ∠C/2. Using the sum of angles in ΔABC (∠A + ∠B + ∠C = 180°), we can substitute and simplify to arrive at the formula above.

Calculating Angle BIC Step-by-Step

We are given the following information for ΔABC:

• The measure of ∠A = 48°.
• The bisectors of ∠B and ∠C meet at point I.

Our goal is to find the measure of ∠BIC.

We will use the established formula for the angle formed by the internal angle bisectors at the incenter:

$$\angle\text{BIC} = 90^\circ + \frac{\angle\text{A}}{2}$$

Now, substitute the given value of ∠A (48°) into the formula:

$$\angle\text{BIC} = 90^\circ + \frac{48^\circ}{2}$$

First, perform the division:

$$\frac{48^\circ}{2} = 24^\circ$$

Next, add this result to 90°:

$$\angle\text{BIC} = 90^\circ + 24^\circ$$

Finally, calculate the sum:

$$\angle\text{BIC} = 114^\circ$$

Angle BIC Calculation Summary

Based on the geometric properties of triangle angle bisectors, the measure of ∠BIC, where I is the incenter formed by the bisectors of ∠B and ∠C in ΔABC with ∠A = 48°, is calculated to be 114°.