Question

[Figure not drawn to scale] The m∠B is 86°, the measure of arc BC is 64°, and the measure of arc CD is 109°. Find the measure of <C. (1 point) Responses 94° 94° 64° 64° 86.5° 86.5° 93.5°

Answer 1

To find the measure of angle \( C \), we need to relate the angles and arcs in the circle.

Given:
- \( m\angle B = 86^\circ \)
- Arc \( BC = 64^\circ \)
- Arc \( CD = 109^\circ \)

Using the inscribed angle theorem, the measure of an angle formed by two points on a circle (like angle \( C \)) is half the measure of the intercepted arc.

Angle \( C \) intercepts the arcs \( BC \) and \( CD \). To find \( m\angle C \), we first need to find the measure of the combined arcs that angle \( C \) intercepts. In this case, angle \( C \) intercepts the arcs \( BC \) and \( CD \).

The measure of the arc intercepted by angle \( C \) is:
\[
m \text{ arc } BC + m \text{ arc } CD = 64^\circ + 109^\circ = 173^\circ
\]

Now, apply the inscribed angle theorem:
\[
m\angle C = \frac{1}{2} \times \text{measure of the intercepted arc}
\]
Thus:
\[
m\angle C = \frac{1}{2} \times 173^\circ = 86.5^\circ
\]

So, the measure of \( \angle C \) is \( 86.5^\circ \).

The correct response is:
**86.5°**