Finding Angle A Using Internal Angle Bisectors

Let's analyze the given problem involving a triangle and its internal angle bisectors. We are given a triangle ΔABC, and the internal bisectors of angles ∠B and ∠C meet at a point, say I. The angle formed at this point I, which is ∠BIC, is given as 132°. We need to find the measure of angle ∠A.

Understanding Internal Angle Bisectors

An internal angle bisector of a triangle divides the angle into two equal parts. The point where the internal angle bisectors of a triangle meet is called the incenter. This point is equidistant from the sides of the triangle.

Formula for Angle Between Internal Bisectors

There is a standard formula that relates the angle formed by the internal bisectors of two angles of a triangle to the third angle of the triangle. If the internal bisectors of ∠B and ∠C of ΔABC meet at I, the angle ∠BIC is given by the formula:

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

Applying the Formula to Solve the Problem

We are given that the angle between the internal bisectors of ∠B and ∠C is 132°. So, we have:

\[ \angle \text{BIC} = 132^\circ \]

Now, we can substitute this value into the formula:

\[ 132^\circ = 90^\circ + \frac{\angle \text{A}}{2} \]

To find the value of ∠A, we need to isolate it. First, subtract 90° from both sides of the equation:

\[ 132^\circ - 90^\circ = \frac{\angle \text{A}}{2} \]

\[ 42^\circ = \frac{\angle \text{A}}{2} \]

Now, multiply both sides by 2 to solve for ∠A:

\[ \angle \text{A} = 42^\circ \times 2 \]

\[ \angle \text{A} = 84^\circ \]

Thus, the value of angle ∠A is 84°.

Step-by-Step Calculation Summary

1. Identify the given information: ∠BIC = 132° (angle between internal bisectors of ∠B and ∠C).
2. Recall the formula: ∠BIC = 90° + ∠A/2.
3. Substitute the known value into the formula: 132° = 90° + ∠A/2.
4. Solve for ∠A:
   • 132° - 90° = ∠A/2
   • 42° = ∠A/2
   • ∠A = 42° * 2
   • ∠A = 84°

Verification

If ∠A = 84°, let's calculate ∠BIC using the formula:

\[ \angle \text{BIC} = 90^\circ + \frac{84^\circ}{2} \]

\[ \angle \text{BIC} = 90^\circ + 42^\circ \]

\[ \angle \text{BIC} = 132^\circ \]

This matches the given information, confirming our calculation is correct.

Revision Table: Key Triangle Angle Formulas

Additional Information: Properties of Incenter and Angle Bisectors

• The incenter is the center of the inscribed circle (incircle) of the triangle.
• The incenter is equidistant from the three sides of the triangle. This distance is the radius of the incircle.
• Every point on an angle bisector is equidistant from the two sides forming the angle.
• Internal angle bisectors always intersect inside the triangle.
• External angle bisectors can intersect inside or outside the triangle, depending on the angles involved.