Certainly! Let's solve both of these questions step-by-step:
### Question 3: How many sides does a regular polygon have if the measure of an exterior angle is 24°?
1. Understanding Exterior Angles:
- For any regular polygon, the sum of all exterior angles is always [tex]\(360^\circ\)[/tex].
- The measure of each exterior angle of a regular polygon is the total angle sum divided by the number of sides [tex]\(n\)[/tex].
2. Given:
- Measure of each exterior angle = [tex]\(24^\circ\)[/tex]
3. Formula:
[tex]\[
\text{Number of sides (n)} = \frac{360^\circ}{\text{Measure of each exterior angle}}
\][/tex]
4. Calculation:
[tex]\[
n = \frac{360^\circ}{24^\circ} = 15
\][/tex]
Answer:
The regular polygon has 15 sides.
### Question 4: How many sides does a regular polygon have if each of its interior angles is 165°?
1. Understanding Interior Angles:
- For any regular polygon, the interior and exterior angles are supplementary, meaning they add up to [tex]\(180^\circ\)[/tex].
- If each interior angle is [tex]\(165^\circ\)[/tex], then each exterior angle can be calculated as:
[tex]\[
\text{Measure of each exterior angle} = 180^\circ - \text{Measure of each interior angle}
\][/tex]
2. Given:
- Measure of each interior angle = [tex]\(165^\circ\)[/tex]
3. Calculation of Exterior Angle:
[tex]\[
\text{Measure of each exterior angle} = 180^\circ - 165^\circ = 15^\circ
\][/tex]
4. Using the same formula as in Question 3:
[tex]\[
n = \frac{360^\circ}{\text{Measure of each exterior angle}}
\][/tex]
5. Calculation:
[tex]\[
n = \frac{360^\circ}{15^\circ} = 24
\][/tex]
Answer:
The regular polygon has 24 sides.
Thus, the regular polygons have:
1. 15 sides if the exterior angle is [tex]\(24^\circ\)[/tex].
2. 24 sides if each interior angle is [tex]\(165^\circ\)[/tex].
