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To solve this problem, we need to determine which lines are parallel in the regular pentagonal prism shown in the image.

A pentagonal prism consists of two congruent pentagons (the top pentagon and the bottom pentagon) connected by vertical edges. In this case, the top pentagon is formed by points \( A, B, C, D, E \), and the bottom pentagon is formed by points \( F, G, H, I, J \) (corresponding to the points in the prism).

Here’s how we identify parallel lines:

1. **Lines within the same pentagon** (top or bottom) that correspond to opposite edges of the pentagon will be parallel. For example, in the top pentagon \( A, B, C, D, E \), opposite edges are parallel.

2. **Vertical lines between the top and bottom pentagon** will also be parallel to each other. These include segments such as \( HE \), \( GF \), etc.

Now let’s evaluate each option:

1. **\( HE \) and \( GF \)**: These two lines connect the top and bottom pentagons. Since they are vertical edges of the prism, they are parallel.
   
2. **\( FE \) and \( HC \)**: These lines connect points in different pentagons. \( FE \) connects \( F \) and \( E \) (bottom to top), and \( HC \) connects \( H \) and \( C \) (also bottom to top). Since they are both vertical edges, they are parallel.
   
3. **\( AB \) and \( HC \)**: \( AB \) is an edge of the top pentagon, and \( HC \) is a vertical edge. These are not parallel because one is horizontal within a pentagon and the other is vertical between two pentagons.
   
4. **\( CD \) and \( HE \)**: \( CD \) is an edge of the top pentagon, while \( HE \) is a vertical edge between the top and bottom pentagons. These are not parallel because they are in different directions.

So, the correct answers are:

- \( HE \) and \( GF \)
- \( FE \) and \( HC \)

Do you need further details or clarifications?

Here are five related questions for further practice:
1. How do you calculate the total surface area of a pentagonal prism?
2. What are the properties of a regular pentagonal prism in terms of symmetry and parallelism?
3. How can you determine if two skew lines are parallel in a 3D figure?
4. What is the volume formula for a pentagonal prism, and how do you apply it?
5. How can you differentiate between parallel and perpendicular lines in 3D geometry?

**Tip**: When dealing with prisms, always remember that vertical edges between congruent faces are usually parallel.

Which lines are parallel in the regular pentagonal prism?

Understand how to identify parallel lines in a regular pentagonal prism. This problem involves concepts from 3D geometry, including prisms and parallelism, and is suitable for students in Grades 8-10. Learn how vertical and opposite edges in prisms are parallel.

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