Answer:
The sum of the measures of the interior angles of the decagon is 1440°
The measure of each interior angle of the decagon is 144°
The measure of each exterior angle of the decagon is 36°
The sum of the measures of the exterior angles of the decagon is 360°
Step-by-step explanation:
- The rule of the sum of the measures of the interior angles of a polygon is (n - 2) × 180°, where n is the number of its side
- If the polygon is regular, then its sides are equal in length and its angles are equal in measures, then the measure of each interior angle is [tex]\frac{(n-2).180}{n}[/tex]
- The interior angle and the exterior angle at a vertex of a polygon formed a pair of linear angles, which means the sum of their measures is 180°
∵ The decagon is a polygon with 10 sides
∴ n = 10
→ Use the first rule above to find the sum of the measures of its
interior angles
∵ The sum of the measures of the interior angles = (10 - 2) × 180°
∴ The sum of the measures of the interior angles = 8 × 180°
∴ The sum of the measures of the interior angles = 1440°
∵ The decagon is regular
→ Use the 2nd rule above
∴ The measure of each interior angle = [tex]\frac{1440}{10}[/tex]
∴ The measure of each interior angle = 144°
→ Use the 3rd rule above
∵ Measure of interior angle + measure of exterior = 180°
∵ The measure of the interior angle = 144°
∴ 144° + measure of exterior angle = 180°
→ Subtract 144 from both sides
∴ The measure of the exterior angle = 180° - 144°
∴ The measure of the exterior angle = 36°
∵ The number of the exterior angles of the decagon is 10
∵ The measure of each exterior angle is 36°
→ Multiply the number of the sides by the measure of each angle
∴ The sum of the measures of the exterior angles = 36° × 10
∴ The sum of the measures of the exterior angles = 360°
