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This question involves determining the symmetries of a regular pentagon. In such a polygon, symmetries include both rotations and reflections.

1. **Rotations for a regular pentagon:**
   - Since a pentagon has five sides, each rotational symmetry is \( \frac{360^\circ}{5} = 72^\circ \).
   - Valid rotations are \(72^\circ\), \(144^\circ\), \(216^\circ\), \(288^\circ\), and \(360^\circ\) around the center (point \(d\)).

2. **Reflections for a regular pentagon:**
   - A pentagon has five lines of symmetry. Each line can either pass through a vertex and the midpoint of the opposite side or through two opposite vertices.
   - Thus, reflections that go through either vertices or opposite sides are valid symmetries.

Let's analyze each option:

- **Option A:** A rotation of \(72^\circ\) around the center \(d\).  
  **Correct.** This is a valid rotation symmetry for a regular pentagon.

- **Option B:** A reflection across line \(a\), through two vertices.  
  **Correct.** This reflection line is a valid line of symmetry for the pentagon.

- **Option C:** A reflection across line \(b\), through one vertex and the center \(d\).  
  **Correct.** This line also represents a valid symmetry for the pentagon.

- **Option D:** A rotation of \(60^\circ\) around the center \(d\).  
  **Incorrect.** Since the pentagon's rotations are multiples of \(72^\circ\), a \(60^\circ\) rotation does not map the pentagon onto itself.

- **Option E:** A reflection across line \(c\), through one vertex and the midpoint of the opposite side.  
  **Correct.** This reflection is also a symmetry of the pentagon.

### Final Answer
The correct options are:
- **A**, **B**, **C**, and **E**.

Would you like more details on any part of this solution? 

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### Further Questions:
1. How can we determine the lines of symmetry for other regular polygons?
2. What are the possible rotational symmetries of an equilateral triangle?
3. Why is a \(60^\circ\) rotation not valid for a pentagon?
4. How would the answer change if the shape were a hexagon instead of a pentagon?
5. Can reflections alone create all the symmetries of a regular pentagon?

### Tip:
For regular polygons with \( n \) sides, each rotation symmetry is an integer multiple of \( \frac{360^\circ}{n} \).

Given the regular polygon, select all of the rotations and reflections that carry the figure onto itself.

Explore the rotational and reflectional symmetries of a regular pentagon. Learn how to determine valid symmetries, including rotations and reflections across various lines. Ideal for high school geometry students.

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