To determine the angle of rotation at which a regular 18-sided polygon will coincide with its preimage, we need to understand how rotational symmetry works in regular polygons.
1. Understanding Rotational Symmetry: A regular polygon has rotational symmetry, meaning that as you rotate the polygon about its center, there exist certain angles where the polygon looks identical to its original position.
2. Calculating the Angle of Rotation for Symmetry: The angle of rotation \( \theta \) that allows a polygon to coincide with its preimage can be determined by dividing the full rotation (360 degrees) by the number of sides \( n \) of the polygon.
[tex]\[
\theta = \frac{360^\circ}{n}
\][/tex]
3. Applying the Formula: Given that the polygon has 18 sides:
[tex]\[
\theta = \frac{360^\circ}{18}
\][/tex]
4. Final Calculation: Performing the division gives:
[tex]\[
\frac{360^\circ}{18} = 20^\circ
\][/tex]
Therefore, the angle of rotation for which an 18-sided regular polygon will coincide with its preimage is:
D. 20°
