To find the measure of angle \( C \), we can use the property of angles formed by two chords that intersect at a point inside a circle. The measure of an angle formed by two chords intersecting inside a circle is equal to half the sum of the measures of the arcs intercepted by the angle and its opposite angle.
Given:
- Measure of angle \( B = 93^\circ \)
- Measure of arc \( BC = 72^\circ \)
- Measure of arc \( CD = 122^\circ \)
From the problem statement, we assume:
- Arc \( BD \) is intercepted by angle \( B \)
- Arc \( BD \) is made up of arcs \( BC \) and \( CD \)
To find the measure of angle \( C \), we first need to determine the measure of arc \( BD \):
\[ \text{Measure of arc } BD = \text{Measure of arc } BC + \text{Measure of arc } CD = 72^\circ + 122^\circ = 194^\circ \]
Now, we can calculate the measure of angle \( C \) using the formula for the measure of the angle formed by two chords:
\[ \text{Measure of angle } C = \frac{1}{2} \left(\text{Measure of arc } BD - \text{Measure of arc } BC\right) \]
Calculating the measure of angle \( C \):
\[ \text{Measure of angle } C = \frac{1}{2} \left(194^\circ - 72^\circ\right) = \frac{1}{2} \cdot 122^\circ = 61^\circ \]
Therefore, the measure of angle \( C \) is \( \boxed{61^\circ} \).
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