Complete parts a and b below.
a. How many reflection symmetries does an eight dash pointed star​ have? How many rotational symmetries does an eight dash pointed star​ have?
b. How many reflection symmetries does a thirteen dash pointed star​ have? How many rotational symmetries does a thirteen dash pointed star​ have?
a.  b.
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Part 1
The eight dash pointed star has

8 reflection symmetries and

rotational symmetries.

To solve this, let's approach the problem in two parts: one for reflection symmetries and one for rotational symmetries.

### Part A: Eight-pointed star
#### Reflection Symmetries:
An eight-pointed star is symmetric, meaning that it can be divided into equal parts through lines passing through the center. Since it has 8 points, there are 8 lines of symmetry, one passing through each point and the center.

Thus, the eight-pointed star has **8 reflection symmetries**.

#### Rotational Symmetries:
For rotational symmetry, we need to determine how many distinct ways the star can be rotated and still look the same. Since the star has 8 points, it will have rotational symmetry every \( \frac{360^\circ}{8} = 45^\circ \). This means the star can be rotated by multiples of 45° and still appear identical. The possible rotations are 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

Thus, the eight-pointed star has **8 rotational symmetries**.

### Part B: Thirteen-pointed star
#### Reflection Symmetries:
For a thirteen-pointed star, we follow the same logic for reflection symmetry. A regular star with 13 points will have reflection symmetry along lines passing through each point and the center. Therefore, there are **13 reflection symmetries**.

#### Rotational Symmetries:
To find the rotational symmetries, we divide 360° by 13. This gives us \( \frac{360^\circ}{13} \approx 27.69^\circ \). Thus, the star can be rotated by multiples of this angle and still appear the same. These rotations include 0°, approximately 27.69°, approximately 55.38°, and so on.

Thus, the thirteen-pointed star has **13 rotational symmetries**.

### Summary of Results:
- **For the eight-pointed star:**
  - Reflection symmetries: 8
  - Rotational symmetries: 8
- **For the thirteen-pointed star:**
  - Reflection symmetries: 13
  - Rotational symmetries: 13

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Would you like further details or have any additional questions?

Here are some related questions:
1. What is the formula for calculating rotational symmetries for any \( n \)-pointed star?
2. How does symmetry change with irregular stars?
3. How can we visualize the lines of reflection in different stars?
4. Can rotational and reflection symmetries be combined in other types of polygons?
5. How do symmetries of stars relate to group theory in mathematics?

**Tip:** Symmetry in stars is closely related to the symmetry in regular polygons, with similar principles governing both reflection and rotational symmetries.

Complete parts a and b below.
a. How many reflection symmetries does an eight-pointed star have? How many rotational symmetries does an eight-pointed star have?
b. How many reflection symmetries does a thirteen-pointed star have? How many rotational symmetries does a thirteen-pointed star have?

Explore how to calculate reflection and rotational symmetries of eight-pointed and thirteen-pointed stars. This problem covers geometric concepts such as symmetry, rotational angles, and reflection lines, providing a detailed solution. Suitable for students in Grades 8-10.

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