In a triangle ΔABC, the internal angle bisectors of angles ∠B and ∠C meet at a point, let's call it O. This point O is known as the incenter of the triangle. The incenter is the center of the inscribed circle that touches all three sides of the triangle internally.

Understanding the Angle Formed by Angle Bisectors

There is a specific relationship between the angle formed by the intersection of two internal angle bisectors and the third angle of the triangle. If the internal bisectors of ∠B and ∠C of ΔABC intersect at O, the measure of ∠BOC can be calculated using the formula:

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

Calculating the Measure of ∠BOC

We are given that in ΔABC, ∠A = 66°. We can use the formula mentioned above to find the measure of ∠BOC.

Substitute the value of ∠A into the formula:

$\angle \text{BOC} = 90^\circ + \frac{66^\circ}{2}$

First, calculate half of ∠A:

$\frac{66^\circ}{2} = 33^\circ$

Now, add this value to 90°:

$\angle \text{BOC} = 90^\circ + 33^\circ$

$\angle \text{BOC} = 123^\circ$

So, the measure of the angle ∠BOC, formed by the intersection of the internal bisectors of ∠B and ∠C, is 123°.

Final Answer

The measure of ∠BOC is 123°.