To find the sum of the measures of the interior angles of a polygon, we can use a specific formula. The formula to calculate the sum of the interior angles of a polygon is:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
where [tex]\( n \)[/tex] is the number of sides of the polygon.
In this problem, the polygon has 13 sides. So we substitute [tex]\( n = 13 \)[/tex] into the formula:
[tex]\[ \text{Sum of interior angles} = (13 - 2) \times 180^\circ \][/tex]
[tex]\[ \text{Sum of interior angles} = 11 \times 180^\circ \][/tex]
[tex]\[ \text{Sum of interior angles} = 1980^\circ \][/tex]
Therefore, the sum of the measures of the interior angles of a 13-sided polygon is [tex]\( \boxed{1980^\circ} \)[/tex]. The correct answer is C.
