To find the measure of angle \(C\) in triangle \(ABC\) given sides \(a = 46\) m, \(b = 30\) m, and \(c = 20\) m, we can use the Law of Cosines, which states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Rearranging this to solve for \(\cos(C)\):
\[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ \cos(C) = \frac{46^2 + 30^2 - 20^2}{2 \cdot 46 \cdot 30} \]
Calculating the squares:
\[ 46^2 = 2116, \quad 30^2 = 900, \quad 20^2 = 400 \]
Now substituting these values:
\[ \cos(C) = \frac{2116 + 900 - 400}{2 \cdot 46 \cdot 30} \]
Calculating the numerator:
\[ 2116 + 900 - 400 = 2616 \]
Calculating the denominator:
\[ 2 \cdot 46 \cdot 30 = 2760 \]
Now compute \(\cos(C)\):
\[ \cos(C) = \frac{2616}{2760} \approx 0.948 \]
Now, finding \(C\):
\[ C = \cos^{-1}(0.948) \]
Calculating:
\[ C \approx 18.9^\circ \]
When rounded to the nearest degree, we find:
\[ C \approx 19^\circ \]
Thus, the measure of \(\angle C\) is 19 degrees.