https://file.aiquickdraw.com/mathbot/file/1730324051065oizpo9a1.jpg

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 \( \frac{360°}{n} \).

Which of the following transformations carry this regular polygon onto itself?

Understand which transformations can carry a regular pentagon onto itself, including rotations by 72° and reflections. This problem explores symmetry concepts and applies rotational and reflectional symmetry for regular polygons, suitable for students in Grades 8-10.

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