The question asks which transformations will carry a regular pentagon onto itself. A regular pentagon has rotational symmetry and reflectional symmetry.

1. Rotation of 45° counterclockwise: This will not map the pentagon onto itself, as a pentagon's internal rotational symmetry requires rotations by multiples of 72° (360° divided by 5).

2. Rotation of 72° counterclockwise: This is correct because rotating a regular pentagon by 72° will map it onto itself, as 72° is the smallest angle of rotation that preserves the pentagon’s symmetry.

3. Rotation of 72° clockwise: This is also correct, for the same reason as above.

4. Reflection across line $l$: If $l$ is a line of symmetry (which appears to be the case in the image), reflecting across it will map the pentagon onto itself.

Correct transformations:

• Rotation of 72° counterclockwise
• Rotation of 72° clockwise
• Reflection across line $l$

Would you like further details on symmetry transformations?

Related Questions

1. What is the minimum angle of rotation that keeps a regular hexagon symmetric?
2. How many lines of symmetry does a regular pentagon have?
3. Why doesn't a 45° rotation work for a pentagon?
4. What transformations keep a regular hexagon symmetric?
5. How do we calculate the smallest rotational symmetry angle for any regular polygon?

Tip

For a regular polygon with $n$ sides, the minimum angle of rotation to map it onto itself is $\frac{360°}{n}$.