To determine the number of sides (n) of a regular polygon given its interior angle, we use the formula for interior angles of regular polygons:
[tex]\[
\text{Interior Angle} = \frac{(n-2) \cdot 180^\circ}{n}
\][/tex]
First, we rearrange this formula to solve for [tex]\( n \)[/tex]:
1. Multiply both sides by [tex]\( n \)[/tex]:
[tex]\[
\text{Interior Angle} \cdot n = (n-2) \cdot 180^\circ
\][/tex]
2. Distribute:
[tex]\[
\text{Interior Angle} \cdot n = 180n - 360
\][/tex]
3. Move all terms involving [tex]\( n \)[/tex] to one side:
[tex]\[
\text{Interior Angle} \cdot n - 180n = -360
\][/tex]
4. Factor out [tex]\( n \)[/tex]:
[tex]\[
n (\text{Interior Angle} - 180) = -360
\][/tex]
5. Divide by the coefficient of [tex]\( n \)[/tex]:
[tex]\[
n = \frac{360}{180 - \text{Interior Angle}}
\][/tex]
Now we will apply this formula to each given interior angle.
### (a) 140°
1. Substitute 140° for the interior angle:
[tex]\[
n = \frac{360}{180 - 140}
\][/tex]
2. Calculate the difference inside the denominator:
[tex]\[
n = \frac{360}{40}
\][/tex]
3. Divide:
[tex]\[
n = 9.0
\][/tex]
Hence, a regular polygon with an interior angle of 140° has 9 sides.
### (b) 108°
1. Substitute 108° for the interior angle:
[tex]\[
n = \frac{360}{180 - 108}
\][/tex]
2. Calculate the difference inside the denominator:
[tex]\[
n = \frac{360}{72}
\][/tex]
3. Divide:
[tex]\[
n = 5.0
\][/tex]
Hence, a regular polygon with an interior angle of 108° has 5 sides.
### (c) 144°
1. Substitute 144° for the interior angle:
[tex]\[
n = \frac{360}{180 - 144}
\][/tex]
2. Calculate the difference inside the denominator:
[tex]\[
n = \frac{360}{36}
\][/tex]
3. Divide:
[tex]\[
n = 10.0
\][/tex]
Hence, a regular polygon with an interior angle of 144° has 10 sides.
### (d) 135°
1. Substitute 135° for the interior angle:
[tex]\[
n = \frac{360}{180 - 135}
\][/tex]
2. Calculate the difference inside the denominator:
[tex]\[
n = \frac{360}{45}
\][/tex]
3. Divide:
[tex]\[
n = 8.0
\][/tex]
Hence, a regular polygon with an interior angle of 135° has 8 sides.
To summarize:
- (a) 140° gives a polygon with 9 sides.
- (b) 108° gives a polygon with 5 sides.
- (c) 144° gives a polygon with 10 sides.
- (d) 135° gives a polygon with 8 sides.
