1. **State the problem:** Determine which given transformations carry a regular polygon (an equilateral triangle in this case) onto itself. 2. **Analyze each transformation:** - The polygon is a regular equilateral triangle. - It has rotational symmetries at multiples of $\frac{360^\circ}{3} = 120^\circ$. - It has reflection symmetries across lines through a vertex and the midpoint of the opposite side. 3. **Check rotations:** - Rotation of $36^\circ$: Since $36$ is not a multiple of $120$, this rotation does not map the triangle onto itself. - Rotation of $120^\circ$: This equals one third of a full rotation, so it does carry the triangle onto itself. - Rotation of $90^\circ$: Since $90$ is not a multiple of $120$, it does not map the triangle onto itself. 4. **Check reflection across $\ell$:** - The line $\ell$ passes horizontally through the center (centroid) of the triangle. - For an equilateral triangle, reflection symmetries occur through lines connecting a vertex and the midpoint of the opposite side. - A horizontal line through the centroid is not such a line of symmetry. - Thus, reflection across $\ell$ does not map the triangle onto itself. **Final answer:** Only the rotation of $120^\circ$ counterclockwise carries this regular polygon onto itself.