Final answer:
An equilateral triangle has rotation symmetry that can map any of its vertices onto any other because all sides and angles are identical, allowing for a 120-degree or 240-degree rotation around the centroid.
Step-by-step explanation:
If a triangle has rotation symmetry that can take any of its vertices to any of its other vertices, the only type of triangle that fits this description is an equilateral triangle. An equilateral triangle has all three sides of the same length and all three angles equal to 60 degrees. This allows rotation symmetry of 120 degrees around the centroid (center of mass) of the triangle. Rotation by 120 degrees or 240 degrees will always map the vertices onto each other, maintaining the shape and orientation of the triangle.
Neither an isosceles triangle, which has only two identical sides and angles, nor a scalene triangle, which has all sides and angles different, can offer this kind of symmetry. Only in an equilateral triangle, every vertex is indistinguishable from the others after rotation, showing full rotational symmetry where each vertex can map onto any other vertex.