To determine the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we can use a well-known formula in geometry. Here’s the step-by-step explanation:
1. Understanding the Problem:
- We need to find a formula that gives us the total sum of the interior angles of a polygon based on the number of its sides.
2. Concept of Polygon Interior Angles:
- A polygon with [tex]\( n \)[/tex] sides (an [tex]\( n \)[/tex]-gon) can be divided into [tex]\( n - 2 \)[/tex] triangles.
- This is because any polygon can be triangulated into [tex]\( n - 2 \)[/tex] non-overlapping triangles.
3. Sum of Angles in Triangles:
- We know the sum of the interior angles of a triangle is [tex]\( 180^\circ \)[/tex].
4. Relating Triangles to the Polygon:
- Since the polygon can be divided into [tex]\( n - 2 \)[/tex] triangles, the sum of the interior angles of the polygon is the sum of the interior angles of these [tex]\( n - 2 \)[/tex] triangles.
5. Calculation:
- Each triangle contributes [tex]\( 180^\circ \)[/tex] to the sum.
- Therefore, the sum is [tex]\( (n - 2) \times 180^\circ \)[/tex].
6. Conclusion:
- The sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by the formula [tex]\( (n-2) \times 180^\circ \)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{(n-2) 180^\circ}
\][/tex]
