To determine the angle measures through which an equilateral triangle can be rotated to map onto itself, we need to consider the symmetry of the triangle. An equilateral triangle has three rotational symmetries:
- 60° - Rotating the triangle by 60° moves each vertex to the position of the next vertex.
- 120° - Rotating the triangle by 120° further moves the vertices around the triangle, also mapping it onto itself.
- 180° - Rotating the triangle by 180° places each vertex into the position of the vertex directly opposite it.
Furthermore, while 240° is essentially the same as rotating 120° (since \(240° = 360° - 120°\)), it is still a valid rotation that maps the triangle onto itself.
So the correct answers are:
- 60°
- 120°
- 180°
- 240°
90° is not a correct answer, as it does not map the triangle onto itself.
Thus, the selected correct responses are: 60°, 120°, 180°, 240°.