Answer :
To solve for the number of sides of a regular polygon given its interior angle, we use the relationship between the interior angle and the number of sides of the polygon.
The formula for the measure of each interior angle of a regular [tex]\( n \)[/tex]-sided polygon is:
[tex]\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
We need to find [tex]\( n \)[/tex], the number of sides, for each given interior angle.
We can manipulate the formula to solve for [tex]\( n \)[/tex] as follows:
[tex]\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
[tex]\[ \text{Interior Angle} \times n = (n-2) \times 180^\circ \][/tex]
[tex]\[ \text{Interior Angle} \times n = 180n - 360 \][/tex]
[tex]\[ n \times \text{Interior Angle} - 180n = -360 \][/tex]
[tex]\[ n(\text{Interior Angle} - 180) = -360 \][/tex]
[tex]\[ n = \frac{360}{180 - \text{Interior Angle}} \][/tex]
Now, let's use this formula to find the number of sides for each given interior angle:
### a) Interior Angle = 135°
[tex]\[ n = \frac{360}{180 - 135} \][/tex]
[tex]\[ n = \frac{360}{45} \][/tex]
[tex]\[ n = 8 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 8 \)[/tex].
### b) Interior Angle = 60°
[tex]\[ n = \frac{360}{180 - 60} \][/tex]
[tex]\[ n = \frac{360}{120} \][/tex]
[tex]\[ n = 3 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 3 \)[/tex].
### c) Interior Angle = 120°
[tex]\[ n = \frac{360}{180 - 120} \][/tex]
[tex]\[ n = \frac{360}{60} \][/tex]
[tex]\[ n = 6 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 6 \)[/tex].
### d) Interior Angle = 140°
[tex]\[ n = \frac{360}{180 - 140} \][/tex]
[tex]\[ n = \frac{360}{40} \][/tex]
[tex]\[ n = 9 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 9 \)[/tex].
So the results are:
- For 135°, the number of sides is [tex]\( 8 \)[/tex].
- For 60°, the number of sides is [tex]\( 3 \)[/tex].
- For 120°, the number of sides is [tex]\( 6 \)[/tex].
- For 140°, the number of sides is [tex]\( 9 \)[/tex].
The formula for the measure of each interior angle of a regular [tex]\( n \)[/tex]-sided polygon is:
[tex]\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
We need to find [tex]\( n \)[/tex], the number of sides, for each given interior angle.
We can manipulate the formula to solve for [tex]\( n \)[/tex] as follows:
[tex]\[ \text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} \][/tex]
[tex]\[ \text{Interior Angle} \times n = (n-2) \times 180^\circ \][/tex]
[tex]\[ \text{Interior Angle} \times n = 180n - 360 \][/tex]
[tex]\[ n \times \text{Interior Angle} - 180n = -360 \][/tex]
[tex]\[ n(\text{Interior Angle} - 180) = -360 \][/tex]
[tex]\[ n = \frac{360}{180 - \text{Interior Angle}} \][/tex]
Now, let's use this formula to find the number of sides for each given interior angle:
### a) Interior Angle = 135°
[tex]\[ n = \frac{360}{180 - 135} \][/tex]
[tex]\[ n = \frac{360}{45} \][/tex]
[tex]\[ n = 8 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 8 \)[/tex].
### b) Interior Angle = 60°
[tex]\[ n = \frac{360}{180 - 60} \][/tex]
[tex]\[ n = \frac{360}{120} \][/tex]
[tex]\[ n = 3 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 3 \)[/tex].
### c) Interior Angle = 120°
[tex]\[ n = \frac{360}{180 - 120} \][/tex]
[tex]\[ n = \frac{360}{60} \][/tex]
[tex]\[ n = 6 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 6 \)[/tex].
### d) Interior Angle = 140°
[tex]\[ n = \frac{360}{180 - 140} \][/tex]
[tex]\[ n = \frac{360}{40} \][/tex]
[tex]\[ n = 9 \][/tex]
Thus, the number of sides of the polygon is [tex]\( 9 \)[/tex].
So the results are:
- For 135°, the number of sides is [tex]\( 8 \)[/tex].
- For 60°, the number of sides is [tex]\( 3 \)[/tex].
- For 120°, the number of sides is [tex]\( 6 \)[/tex].
- For 140°, the number of sides is [tex]\( 9 \)[/tex].
To find the number of sides of a regular polygon based on the measure of its interior angles, we can use the formula:
Interior Angle =
(
(n - 2) × 180
n
where n
represents the number of sides of the polygon.
a) For an interior angle of 135°:
Substitute 135°
into the formula:
(n - 2) × 180
135 =
n
Solve for n :
135n = 180n - 360
45n = 360
n =
360
45
n = 8
Therefore, the polygon has 8 sides.
b) For an interior angle of 60°:
Substitute 60°
into the formula:
60 =
( (n - 2) × 180
n
Solve for n
60n = 180n - 360
120n = 360
n =
360
120
n = 3
Therefore, the polygon has 3 sides, which is a triangle.
c) For an interior angle of 120°:
Substitute 120°
into the formula: 120 = (
(02- 211 180)
Solve for n
120n = 180n - 360
60n = 360
360
n =
60
n = 6
Therefore, the polygon has 6 sides.
d) For an interior angle of 140°:
Substitute 140 A° into the formula:
(n - 2) × 180
140 =
Solve for n
140n = 180n - 360
This equation does not lead to a whole number solution, indicating that there is no reaular polvaon with.
Solve for n
140n = 180n - 360
This equation does not lead to a whole number solution, indicating that there is no regular polygon with interior angles of 140°.
In summary:
a) 8 sides
b) 3 sides (triangle)
c) 6 sides
d) No regular polygon with this interior angle
Interior Angle =
(
(n - 2) × 180
n
where n
represents the number of sides of the polygon.
a) For an interior angle of 135°:
Substitute 135°
into the formula:
(n - 2) × 180
135 =
n
Solve for n :
135n = 180n - 360
45n = 360
n =
360
45
n = 8
Therefore, the polygon has 8 sides.
b) For an interior angle of 60°:
Substitute 60°
into the formula:
60 =
( (n - 2) × 180
n
Solve for n
60n = 180n - 360
120n = 360
n =
360
120
n = 3
Therefore, the polygon has 3 sides, which is a triangle.
c) For an interior angle of 120°:
Substitute 120°
into the formula: 120 = (
(02- 211 180)
Solve for n
120n = 180n - 360
60n = 360
360
n =
60
n = 6
Therefore, the polygon has 6 sides.
d) For an interior angle of 140°:
Substitute 140 A° into the formula:
(n - 2) × 180
140 =
Solve for n
140n = 180n - 360
This equation does not lead to a whole number solution, indicating that there is no reaular polvaon with.
Solve for n
140n = 180n - 360
This equation does not lead to a whole number solution, indicating that there is no regular polygon with interior angles of 140°.
In summary:
a) 8 sides
b) 3 sides (triangle)
c) 6 sides
d) No regular polygon with this interior angle
