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.