To find the measure of angle \( C \) in triangle \( ABC \) with sides \( a = 46 , \text{m} \), \( b = 30 , \text{m} \), and \( c = 20 , \text{m} \), we can use the Law of Cosines. The law states:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
Substituting in the values for \( a \), \( b \), and \( c \):
\[ 20^2 = 46^2 + 30^2 - 2 \cdot 46 \cdot 30 \cdot \cos(C) \]
Calculating the squares:
\[ 400 = 2116 + 900 - 2760 \cos(C) \]
Combining terms:
\[ 400 = 3016 - 2760 \cos(C) \]
Rearranging gives:
\[ 2760 \cos(C) = 3016 - 400 \] \[ 2760 \cos(C) = 2616 \]
Now, dividing both sides by 2760:
\[ \cos(C) = \frac{2616}{2760} \] \[ \cos(C) \approx 0.948 \]
Now, using the inverse cosine function to find angle \( C \):
\[ C = \cos^{-1}(0.948) \]
Calculating that gives:
\[ C \approx 18.2 \text{ degrees} \]
Rounding to the nearest degree:
\[ C \approx 18 \text{ degrees} \]
However, that value is not one of the options provided. The closest available option is 19 degrees. Therefore, the measure of angle \( C \) to the nearest degree is:
19 degrees