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To determine which transformations would carry the regular polygon (an octagon in this case) onto itself, we need to consider its symmetry.

For a regular octagon:
1. **Rotational Symmetry**: The octagon has 8-fold rotational symmetry. This means it can be rotated by multiples of \( \frac{360^\circ}{8} = 45^\circ \) to map onto itself.
2. **Reflectional Symmetry**: The octagon has reflectional symmetry across lines that pass through opposite vertices and also across lines that pass through the midpoints of opposite sides.

Let's analyze each option:
1. **Rotation of 30° counterclockwise**: Not a multiple of 45°, so it does **not** map the octagon onto itself.
2. **Rotation of 40° counterclockwise**: Also not a multiple of 45°, so it does **not** map the octagon onto itself.
3. **Reflection across line \( \ell \)**: If line \( \ell \) passes through vertices or midpoints of opposite sides, it would reflect the octagon onto itself. Therefore, this **could** be a valid transformation.
4. **Rotation of 36° clockwise**: Not a multiple of 45°, so it does **not** map the octagon onto itself.

### Correct Answer
The only transformation that would map the octagon onto itself is the **reflection across line \( \ell \)**, provided \( \ell \) is aligned with the symmetry axes of the octagon.

Would you like more detailed explanations on symmetry properties or have additional questions?

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#### Related Questions
1. What are the possible rotation angles that would map a regular hexagon onto itself?
2. How many lines of symmetry does a regular pentagon have?
3. Why can't rotations of 30° or 36° map a regular polygon with 8 sides onto itself?
4. What transformations map a regular dodecagon onto itself?
5. How can we determine if a line is a symmetry axis for a given polygon?

#### Tip
When working with regular polygons, the smallest rotation that maps the shape onto itself is always \( \frac{360^\circ}{\text{number of sides}} \).

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

Understand which transformations, including specific rotation and reflection, map a regular octagon onto itself. This problem explores geometric symmetry concepts, such as rotational and reflectional symmetry, suitable for students in Grades 7-9.

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