The measure of the supplementary angle of an interior angle of a regular polygon can be determined by first understanding the formula for finding the measure of an interior angle.
For a regular polygon, the formula for calculating the measure of an interior angle (A) is:
A = [(n – 2) × 180°] / n
In this formula, ‘n’ represents the number of sides of the polygon. Once you have calculated the interior angle, the supplementary angle (S) can simply be found using the following relationship:
S = 180° – A
This means that the supplementary angle is what, when added to the interior angle, results in 180°. Therefore, to find the supplementary angle:
- Determine the number of sides of the polygon (n).
- Use the interior angle formula to find A.
- Subtract A from 180° to get the supplementary angle S.
For example, consider a regular pentagon (n = 5). The interior angle would be:
A = [(5 – 2) × 180°] / 5 = [3 × 180°] / 5 = 108°
Then, the supplementary angle would be:
S = 180° – 108° = 72°
Thus, in the case of a regular pentagon, the measure of the supplementary angle of an interior angle is 72°.