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.
For a polygon with [tex]\( n \)[/tex] sides, also known as an [tex]\( n \)[/tex]-gon, the sum of the interior angles is given by:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
Let's break down why this formula makes sense:
1. Understand the concept:
- A polygon can be divided into triangles. For example, a triangle (3 sides) is already the simplest polygon, and its angle sum is [tex]\( 180^\circ \)[/tex].
- A quadrilateral (4 sides) can be divided into two triangles, so the sum of interior angles is [tex]\( 2 \times 180^\circ = 360^\circ \)[/tex].
2. Generalize:
- The general rule is that any [tex]\( n \)[/tex]-sided polygon can be divided into [tex]\( (n - 2) \)[/tex] triangles.
3. Apply the formula to any polygon:
- Since each triangle's angles sum up to [tex]\( 180^\circ \)[/tex], multiplying the number of such triangles, [tex]\( (n-2) \)[/tex], by [tex]\( 180^\circ \)[/tex] gives the total sum of the interior angles for the polygon.
Hence for an [tex]\( n \)[/tex]-sided polygon:
[tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \][/tex]
Given the options:
- A. [tex]\((n-2) \times 360^\circ\)[/tex]
- B. [tex]\((n-3) \times 180^\circ\)[/tex]
- C. [tex]\((n-2) \times 90^\circ\)[/tex]
- D. [tex]\((n-3) \times 3600^\circ\)[/tex]
- E. [tex]\((n-2) \times 180^\circ\)[/tex]
The correct sum of the interior angles is [tex]\((n - 2) \times 180^\circ\)[/tex], which matches option E.
Therefore, the correct answer is:
[tex]\[ \boxed{(n-2) \times 180^\circ} \][/tex]
