To determine if a given angle can be an exterior angle of a regular polygon, we can use the formula for the exterior angle of a regular polygon, which is given by:
\[ \text{Exterior angle} = \frac{360}{n} \]
where \(n\) is the number of sides of the polygon. This means that for a regular polygon, the exterior angle must be a divisor of 360 degrees.
To check each of the provided angles, we'll find \(n\) using the rearranged formula:
\[ n = \frac{360}{\text{Exterior angle}} \]
We will also need to ensure that \(n\) is a positive integer, meaning the exterior angle must be a divisor of 360.
Now let's evaluate each angle:
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90 degrees: \[ n = \frac{360}{90} = 4 \quad \text{(possible, a square)} \]
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80 degrees: \[ n = \frac{360}{80} = 4.5 \quad \text{(not a whole number, not possible)} \]
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75 degrees: \[ n = \frac{360}{75} = 4.8 \quad \text{(not a whole number, not possible)} \]
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30 degrees: \[ n = \frac{360}{30} = 12 \quad \text{(possible, a dodecagon)} \]
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46 degrees: \[ n = \frac{360}{46} \approx 7.826 \quad \text{(not a whole number, not possible)} \]
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36 degrees: \[ n = \frac{360}{36} = 10 \quad \text{(possible, a decagon)} \]
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2 degrees: \[ n = \frac{360}{2} = 180 \quad \text{(possible, an 180-sided polygon)} \]
Summary of Results:
- 90 degrees: Possible (4 sides - square)
- 80 degrees: Not possible
- 75 degrees: Not possible
- 30 degrees: Possible (12 sides - dodecagon)
- 46 degrees: Not possible
- 36 degrees: Possible (10 sides - decagon)
- 2 degrees: Possible (180 sides)
Therefore, the possible measures for the exterior angles along with the number of sides of their respective polygons are:
- 90 degrees: 4 sides
- 30 degrees: 12 sides
- 36 degrees: 10 sides
- 2 degrees: 180 sides
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