To determine the sum of the measures of the interior angles of a polygon with [tex]\( n \)[/tex] sides, we use a well-known formula in geometry. Let's go through the steps to derive this formula and apply it to the given options.
1. Understanding the Polygon and Interior Angles:
- A polygon with [tex]\( n \)[/tex] sides is called an n-gon.
- The sum of the measures of the interior angles of an n-gon can be derived by dividing the polygon into triangles.
2. Dividing the Polygon into Triangles:
- If you connect one vertex of the polygon to all other non-adjacent vertices, you can divide the polygon into [tex]\( n - 2 \)[/tex] triangles. This property helps in developing the formula.
- Each triangle has an interior angle sum of [tex]\( 180^\circ \)[/tex].
3. Summing the Interior Angles of Triangles:
- Therefore, the sum of the interior angles of the polygon is the sum of the interior angles of these [tex]\( n - 2 \)[/tex] triangles.
- Mathematically, this is expressed as: [tex]\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ. \][/tex]
4. Matching with Given Options:
- Option A: [tex]\((n-3) 3600^\circ\)[/tex] does not use the correct components or unit.
- Option B: [tex]\((n-2) 180^\circ\)[/tex] exactly matches our derived formula.
- Option C: [tex]\((n-2) 90^\circ\)[/tex] uses the wrong multiplier.
- Option D: [tex]\((n-2) 360^\circ\)[/tex] uses an incorrect angle.
- Option E: [tex]\((n-3) 180^\circ\)[/tex] uses the incorrect starting term [tex]\((n-3)\)[/tex] instead of [tex]\((n-2)\)[/tex].
Given the correct derivation of the sum of the interior angles formula, the correct choice is:
[tex]\[ \boxed{(n-2) 180^\circ} \][/tex]
Thus, the correct answer is option B.
