To find the measure of angle \( C \) in triangle \( AABC \) using the lengths of the sides \( a = 46 \) m, \( b = 30 \) m, and \( c = 20 \) m, we can use the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
Rearranging to solve for \( \cos(C) \):
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values:
- \( a = 46 \)
- \( b = 30 \)
- \( c = 20 \)
First, calculate \( a^2 \), \( b^2 \), and \( c^2 \):
\[ a^2 = 46^2 = 2116 \] \[ b^2 = 30^2 = 900 \] \[ c^2 = 20^2 = 400 \]
Now plug these into the formula:
\[ \cos(C) = \frac{2116 + 900 - 400}{2 \cdot 46 \cdot 30} \] \[ = \frac{2616}{2760} \] \[ = 0.948 \]
Now, take the arccosine to find \( C \):
\[ C = \cos^{-1}(0.948) \]
Calculating \( C \):
\[ C \approx 18.19^\circ \]
Rounding to the nearest degree, we get:
\[ C \approx 19^\circ \]
Thus, the measure of angle \( C \) to the nearest degree is \( \boxed{19} \) degrees.