To find the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we use a well-known geometric formula. Here’s a detailed step-by-step explanation:
1. Understand the Formula:
The formula to determine the sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is given by:
[tex]\[
\text{Sum of interior angles} = (n - 2) \times 180^\circ
\][/tex]
This formula arises because a polygon can be divided into [tex]\((n - 2)\)[/tex] triangles, and the sum of the interior angles of each triangle is [tex]\(180^\circ\)[/tex].
2. Explanation:
- A polygon with [tex]\( n \)[/tex] sides can be split into [tex]\((n - 2)\)[/tex] triangles by drawing diagonals from one vertex to all other non-adjacent vertices.
- Each triangle formed in this manner will have angles summing to [tex]\(180^\circ\)[/tex].
- Therefore, the sum of all interior angles of the polygon will be [tex]\((n - 2) \times 180^\circ\)[/tex].
3. Identifying the Correct Option:
Let's now compare the given options to our formula for the sum of interior angles:
[tex]\[
\begin{align*}
\text{A.} & \quad (n - 3) \times 3600^\circ \\
\text{B.} & \quad (n - 3) \times 180^\circ \\
\text{C.} & \quad (n - 2) \times 90^\circ \\
\text{D.} & \quad (n - 2) \times 180^\circ \quad (\text{This matches our formula}) \\
\text{E.} & \quad (n - 2) \times 360^\circ \\
\end{align*}
\][/tex]
4. Conclusion:
The correct formula for the sum of the interior angles of a polygon with [tex]\( n \)[/tex] sides is [tex]\((n - 2) \times 180^\circ\)[/tex]. Therefore, the correct answer among the options provided is:
[tex]\[
\boxed{(n - 2) \times 180^\circ}
\][/tex]
Thus, the answer is option D.
