To determine two different degrees of rotation less than 120 degrees but greater than 0 that will turn a regular polygon into itself (meaning it will look the same after the rotation), we need to consider the properties of regular polygons.
A regular polygon can be rotated around its center. The key is that the angle of rotation must be a multiple of the angle formed by dividing 360 degrees by the number of sides of the polygon.
Let's take a regular hexagon (6-sided polygon) as an example:
- The angle for rotation is \( \frac{360^\circ}{6} = 60^\circ \).
- Therefore, two valid rotations greater than 0 and less than 120 degrees would be:
- 60 degrees
- 120 degrees (not allowed, as it is equal to 120)
Thus, for the hexagon, the rotation of only one degree (for example, halfway between 60 and 120) would suffice. However, we need two distinct rotations.
Let's switch to a regular pentagon (5-sided polygon):
- The angle for rotation is \( \frac{360^\circ}{5} = 72^\circ \).
- Therefore, two valid rotations would be:
- 72 degrees
- 144 degrees (not valid)
Thus, for both hexagon and pentagon, we may choose:
- For a hexagon: 60 degrees and another rotation of 120 degrees.
- For a pentagon: 72 degrees and the half-way mark of 36 degrees.
In summary, for distinct angles to rotate a regular polygon and make it look the same, consider:
- Regular Hexagon: Rotations of 60° and 120° (if allowable).
- Regular Pentagon: Rotations of 72° and 36°.
To comply with your condition, you may decide if you would present them under specific polygons, or general angles regardless of the polygon will also apply each rotation derived directly by considering rotational symmetry.
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