The image shows an equilateral triangle with labeled sides, and the problem is asking which transformations carry this regular polygon (triangle) onto itself.

The correct transformations for an equilateral triangle (a regular polygon) are as follows:

1. Rotation of 120° counterclockwise: This will map each vertex of the triangle onto another vertex, thus preserving the shape. Since an equilateral triangle has three vertices, rotating by 120° (which is 360°/3) will map the triangle onto itself.

2. Rotation of 72° counterclockwise: This is incorrect for an equilateral triangle because 72° is associated with a pentagon, not a triangle. A triangle cannot be mapped onto itself with this rotation.

3. Reflection across the marked line of symmetry ($\ell$): A reflection across the line of symmetry will also map the triangle onto itself. Equilateral triangles have three lines of symmetry, one for each vertex, and the line shown in the diagram represents one of those lines.

In summary, the transformations that will map this equilateral triangle onto itself are:

• 120° counterclockwise rotation.
• Reflection across the line $\ell$.

Would you like more details or have any other questions?

Related Questions:

1. What are the other valid rotations for an equilateral triangle?
2. How many lines of symmetry does a regular polygon have, and why?
3. Why does a rotation of 72° work for a regular pentagon but not for a triangle?
4. Can an equilateral triangle be mapped onto itself with any translation?
5. What are the properties of transformations that preserve a shape?

Tip:

Remember that for any regular polygon with $n$ sides, the angle of rotation to map the polygon onto itself is $360^\circ/n$.