Answer :
Sure, let's solve the problem step-by-step for each part:
### Finding the Number of Sides of a Regular Polygon for Given Exterior Angles:
a) Exterior Angle: 24°
For any regular polygon, the sum of the exterior angles always equals 360°. To find the number of sides ([tex]\( n \)[/tex]), we can use the formula:
[tex]\[ n = \frac{360^\circ}{\text{Exterior Angle}} \][/tex]
For an exterior angle of 24°:
[tex]\[ n = \frac{360^\circ}{24^\circ} = 15 \][/tex]
So, the polygon has 15 sides.
b) Exterior Angle: 60°
Using the same formula:
For an exterior angle of 60°:
[tex]\[ n = \frac{360^\circ}{60^\circ} = 6 \][/tex]
So, the polygon has 6 sides.
### Finding the Size of Each Interior Angle for Given Exterior Angles:
The interior angle ([tex]\(\text{Interior Angle}\)[/tex]) of a regular polygon can be found using the relationship:
[tex]\[ \text{Interior Angle} = 180^\circ - \text{Exterior Angle} \][/tex]
c) Exterior Angle: 72°
For an exterior angle of 72°, the interior angle is:
[tex]\[ \text{Interior Angle} = 180^\circ - 72^\circ = 108^\circ \][/tex]
d) Exterior Angle: 120°
For an exterior angle of 120°, the interior angle is:
[tex]\[ \text{Interior Angle} = 180^\circ - 120^\circ = 60^\circ \][/tex]
### Summary of Answers:
- Number of sides:
- For exterior angle 24°: 15 sides
- For exterior angle 60°: 6 sides
- Interior angles:
- For exterior angle 72°: 108°
- For exterior angle 120°: 60°
To conclude, the number of sides and the corresponding interior angles for the given exterior angles are as follows:
- [tex]\(24^\circ\)[/tex] : 15 sides
- [tex]\(60^\circ\)[/tex] : 6 sides
- [tex]\(72^\circ\)[/tex] : [tex]\(\text{Interior Angle} = 108^\circ\)[/tex]
- [tex]\(120^\circ\)[/tex] : [tex]\(\text{Interior Angle} = 60^\circ\)[/tex]
### Finding the Number of Sides of a Regular Polygon for Given Exterior Angles:
a) Exterior Angle: 24°
For any regular polygon, the sum of the exterior angles always equals 360°. To find the number of sides ([tex]\( n \)[/tex]), we can use the formula:
[tex]\[ n = \frac{360^\circ}{\text{Exterior Angle}} \][/tex]
For an exterior angle of 24°:
[tex]\[ n = \frac{360^\circ}{24^\circ} = 15 \][/tex]
So, the polygon has 15 sides.
b) Exterior Angle: 60°
Using the same formula:
For an exterior angle of 60°:
[tex]\[ n = \frac{360^\circ}{60^\circ} = 6 \][/tex]
So, the polygon has 6 sides.
### Finding the Size of Each Interior Angle for Given Exterior Angles:
The interior angle ([tex]\(\text{Interior Angle}\)[/tex]) of a regular polygon can be found using the relationship:
[tex]\[ \text{Interior Angle} = 180^\circ - \text{Exterior Angle} \][/tex]
c) Exterior Angle: 72°
For an exterior angle of 72°, the interior angle is:
[tex]\[ \text{Interior Angle} = 180^\circ - 72^\circ = 108^\circ \][/tex]
d) Exterior Angle: 120°
For an exterior angle of 120°, the interior angle is:
[tex]\[ \text{Interior Angle} = 180^\circ - 120^\circ = 60^\circ \][/tex]
### Summary of Answers:
- Number of sides:
- For exterior angle 24°: 15 sides
- For exterior angle 60°: 6 sides
- Interior angles:
- For exterior angle 72°: 108°
- For exterior angle 120°: 60°
To conclude, the number of sides and the corresponding interior angles for the given exterior angles are as follows:
- [tex]\(24^\circ\)[/tex] : 15 sides
- [tex]\(60^\circ\)[/tex] : 6 sides
- [tex]\(72^\circ\)[/tex] : [tex]\(\text{Interior Angle} = 108^\circ\)[/tex]
- [tex]\(120^\circ\)[/tex] : [tex]\(\text{Interior Angle} = 60^\circ\)[/tex]
