A regular hexagon is rotated about its center. Which degree measure will carry the regular hexagon onto itself? 1) $$ 45^{\circ} $$

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To determine which degree measures will carry a regular hexagon onto itself when rotated about its center, we need to consider the symmetry of the hexagon.

A regular hexagon has 6 sides and is symmetric. The angles of rotation that will map the hexagon onto itself are those that correspond to the angles between its vertices.

1. The full rotation of the hexagon is $$360^{\circ}$$.
2. Since there are 6 vertices, the angle of rotation that maps the hexagon onto itself can be calculated as:

$$
\text{Angle of rotation} = \frac{360^{\circ}}{6} = 60^{\circ}
$$

This means that rotating the hexagon by $$60^{\circ}$$, $$120^{\circ}$$, $$180^{\circ}$$, $$240^{\circ}$$, $$300^{\circ}$$, or $$360^{\circ}$$ will carry the hexagon onto itself.

Now, let's evaluate the given option:

- $$45^{\circ}$$ is not one of the angles that will map the hexagon onto itself.

Thus, the answer is that $$45^{\circ}$$ will not carry the regular hexagon onto itself. The correct angles are $$60^{\circ}$$, $$120^{\circ}$$, $$180^{\circ}$$, $$240^{\circ}$$, $$300^{\circ}$$, and $$360^{\circ}$$.
The regular hexagon will be carried onto itself when rotated by $$60^{\circ}$$, $$120^{\circ}$$, $$180^{\circ}$$, $$240^{\circ}$$, $$300^{\circ}$$, or $$360^{\circ}$$. Therefore, $$45^{\circ}$$ does not carry the hexagon onto itself.

A regular hexagon is rotated about its center. Which degree measure will carry the regular hexagon onto itself? 1) \( 45^{\circ} \)

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