Final answer:
A regular decagon maps onto itself through rotations of 36° and 72°, as these are multiples of the angle that corresponds to the decagon's rotational symmetry. The correct option is A.
Step-by-step explanation:
A regular decagon is a 10-sided polygon where all sides and angles are equal. When we discuss a figure 'mapping onto itself,' we are referring to a concept known as rotational symmetry. This concept means that the figure looks the same after a certain amount of rotation less than a full turn (360°).
To find the angle measures through which a regular decagon can be rotated to map onto itself, we divide 360° by the number of sides the decagon has. Since a decagon has 10 sides, we calculate 360° / 10 = 36°. Therefore, every 36° of rotation will map the decagon onto itself. This is the smallest angle of rotational symmetry for a regular decagon.
As the decagon has 10 sides, it can be mapped onto itself at multiple of these angles, meaning at 36°, 72° (2 x 36°), 108° (3 x 36°), and so on, up to 324° (9 x 36°). Considering the answer choices provided in the question, the only angle measures that are multiples of 36° are 36° and 72°. Therefore, these are the angles through which the regular decagon can be rotated to map onto itself.
Converting between radians, revolutions, and degrees is essential for understanding rotation. One revolution is equivalent to 2п radians or 360°, and hence the angle of rotation of one revolution is 2п radians. To convert radians to degrees, we use the fact that one radian is approximately 57.3°.