To find the measure of angle \( C \) using the Law of Sines, we start with the formula:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Given:
- \( b = 13 \) inches
- \( c = 6 \) inches
- \( \angle B = 94^\circ \)
We can find \( \sin B \):
\[ \sin B = \sin(94^\circ) \]
Now, we can write the ratio for \( b \) and \( c \):
\[ \frac{b}{\sin B} = \frac{c}{\sin C} \]
Substituting the known values:
\[ \frac{13}{\sin(94^\circ)} = \frac{6}{\sin C} \]
Now isolate \( \sin C \):
\[ \sin C = \frac{6 \cdot \sin(94^\circ)}{13} \]
Calculating \( \sin(94^\circ) \):
\[ \sin(94^\circ) \approx 0.998 \]
Now substituting back in:
\[ \sin C = \frac{6 \cdot 0.998}{13} \approx \frac{5.988}{13} \approx 0.4606 \]
Next, we find \( C \) by taking the inverse sine:
\[ C \approx \sin^{-1}(0.4606) \]
Calculating \( C \):
\[ C \approx 27.4^\circ \]
Rounding to the nearest degree:
\[ C \approx 27^\circ \]
Thus, the measure of angle \( C \) is \( 27 \) degrees.
The response is \( \boxed{27} \) degrees.