Question

Provide two different degrees of rotation less than 120 degrees but greater than 0 that will turn a regular octagon onto itself.

Answer 1

A regular octagon has rotational symmetries that correspond to multiples of \( \frac{360^\circ}{8} = 45^\circ \). To find degrees of rotation that turn the octagon onto itself and are within the specified range (greater than 0 and less than 120 degrees), we can consider the multiples of 45 degrees.

The possible rotations of the octagon are:

1. \( 45^\circ \)
2. \( 90^\circ \)
3. \( 135^\circ \) (which is greater than 120 degrees and therefore not valid)
4. \( 180^\circ \) (also not valid since it's greater than 120 degrees)

From this list, the two valid degrees of rotation that turn a regular octagon onto itself while meeting the condition of being greater than 0 and less than 120 degrees are:

- **45 degrees**
- **90 degrees**

Thus, the two degrees of rotation are \( 45^\circ \) and \( 90^\circ \).