To find the angle \( \angle C \) in triangle \( ABC \) given sides \( a = 46 \) m, \( b = 30 \) m, and \( c = 20 \) m, we can use the Cosine Rule, which states:
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values of the sides into the formula:
\[ a = 46, \quad b = 30, \quad c = 20 \]
Calculating \( a^2 \), \( b^2 \), and \( c^2 \):
\[ a^2 = 46^2 = 2116 \] \[ b^2 = 30^2 = 900 \] \[ c^2 = 20^2 = 400 \]
Now substituting these values into the cosine formula:
\[ \cos(C) = \frac{2116 + 900 - 400}{2 \times 46 \times 30} \] \[ = \frac{2616}{2760} \] \[ = 0.94886 \]
Next, we compute \( C \) by taking the arccosine of \( 0.94886 \):
\[ C = \cos^{-1}(0.94886) \]
Using a calculator to find \( C \):
\[ C \approx 18.9^\circ \]
Rounding to the nearest degree, we find \( C \approx 19^\circ \).
Thus, the measure of \( \angle C \) to the nearest degree is:
19 degrees.