Identify parallel, perpendicular, and intersecting lines in the given diagrams.

Problem Analysis:

The task involves identifying the relationships between pairs of lines in a given geometric plane. The relationships to identify are:
1. Parallel lines: Lines that never intersect and are always the same distance apart.
2. Perpendicular lines: Lines that intersect at a 90-degree angle.
3. Intersecting lines: Lines that cross each other at any angle (not necessarily 90 degrees).

We need to analyze the image provided and determine the relationship between each pair of lines listed.

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Step-by-Step Solution:

#### Given Image:
The image shows a geometric plane with several labeled lines:
- Line \( \ell \)
- Line \( m \)
- Line \( n \)
- Line \( p \)
- Line \( q \)

From the diagram, we can observe the following:
1. Line \( \ell \): A horizontal line.
2. Line \( m \): A vertical line.
3. Line \( n \): A diagonal line sloping upward from left to right.
4. Line \( p \): A diagonal line sloping downward from left to right.
5. Line \( q \): Another horizontal line parallel to \( \ell \).

#### Analysis of Each Pair:

1. \( \ell \) and \( m \):
- Line \( \ell \) is horizontal.
- Line \( m \) is vertical.
- These lines intersect at a 90-degree angle.
- Conclusion: \( \ell \) and \( m \) are perpendicular.

2. \( \ell \) and \( n \):
- Line \( \ell \) is horizontal.
- Line \( n \) is diagonal and slopes upward.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( \ell \) and \( n \) are intersecting.

3. \( \ell \) and \( p \):
- Line \( \ell \) is horizontal.
- Line \( p \) is diagonal and slopes downward.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( \ell \) and \( p \) are intersecting.

4. \( \ell \) and \( q \):
- Line \( \ell \) is horizontal.
- Line \( q \) is also horizontal and does not intersect \( \ell \).
- These lines are always the same distance apart.
- Conclusion: \( \ell \) and \( q \) are parallel.

5. \( m \) and \( n \):
- Line \( m \) is vertical.
- Line \( n \) is diagonal and slopes upward.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( m \) and \( n \) are intersecting.

6. \( m \) and \( p \):
- Line \( m \) is vertical.
- Line \( p \) is diagonal and slopes downward.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( m \) and \( p \) are intersecting.

7. \( n \) and \( p \):
- Line \( n \) is diagonal and slopes upward.
- Line \( p \) is diagonal and slopes downward.
- These lines intersect at a point.
- Conclusion: \( n \) and \( p \) are intersecting.

8. \( n \) and \( q \):
- Line \( n \) is diagonal and slopes upward.
- Line \( q \) is horizontal.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( n \) and \( q \) are intersecting.

9. \( p \) and \( q \):
- Line \( p \) is diagonal and slopes downward.
- Line \( q \) is horizontal.
- These lines intersect at an angle other than 90 degrees.
- Conclusion: \( p \) and \( q \) are intersecting.

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Final Answer:

\[
\begin{array}{|c|c|}
\hline
\text{Lines} & \text{Relationship} \\
\hline
\ell \text{ and } m & \text{Perpendicular} \\
\hline
\ell \text{ and } n & \text{Intersecting} \\
\hline
\ell \text{ and } p & \text{Intersecting} \\
\hline
\ell \text{ and } q & \text{Parallel} \\
\hline
m \text{ and } n & \text{Intersecting} \\
\hline
m \text{ and } p & \text{Intersecting} \\
\hline
n \text{ and } p & \text{Intersecting} \\
\hline
n \text{ and } q & \text{Intersecting} \\
\hline
p \text{ and } q & \text{Intersecting} \\
\hline
\end{array}
\]

\boxed{
\begin{array}{|c|c|}
\hline
\text{Lines} & \text{Relationship} \\
\hline
\ell \text{ and } m & \text{Perpendicular} \\
\hline
\ell \text{ and } n & \text{Intersecting} \\
\hline
\ell \text{ and } p & \text{Intersecting} \\
\hline
\ell \text{ and } q & \text{Parallel} \\
\hline
m \text{ and } n & \text{Intersecting} \\
\hline
m \text{ and } p & \text{Intersecting} \\
\hline
n \text{ and } p & \text{Intersecting} \\
\hline
n \text{ and } q & \text{Intersecting} \\
\hline
p \text{ and } q & \text{Intersecting} \\
\hline
\end{array}
}