To determine the correct formula for the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we can derive it step-by-step.
1. Understanding the Interior Angles of Polygons:
- A polygon's interior angles collectively add up to a certain measure depending on the number of sides.
- For a polygon with 3 sides (a triangle), the sum of the interior angles is always [tex]\( 180^\circ \)[/tex].
- For a polygon with 4 sides (a quadrilateral), the sum of the interior angles is [tex]\( 360^\circ \)[/tex].
- Generally, for a polygon with [tex]\( n \)[/tex] sides, the pattern can be found.
2. Derive a General Formula:
- One way to derive this pattern is to divide a polygon into triangles. For a polygon with [tex]\( n \)[/tex] sides, you can draw [tex]\( n-2 \)[/tex] triangles within it. This can be done by drawing diagonals from one vertex to all other non-adjacent vertices.
- Each triangle has an interior angle sum of [tex]\( 180^\circ \)[/tex].
- Therefore, the sum of the interior angles for a polygon can be calculated as [tex]\( (n-2) \times 180^\circ \)[/tex].
3. Verification:
- For [tex]\( n = 3 \)[/tex]: [tex]\( (3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ \)[/tex] (Correct for triangle)
- For [tex]\( n = 4 \)[/tex]: [tex]\( (4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \)[/tex] (Correct for quadrilateral)
Given the derived formula, the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides is [tex]\( (n-2) \times 180^\circ \)[/tex].
Thus, the correct choice among the provided options is:
[tex]\[ \text{C. } (n-2) \times 180^\circ \][/tex]
