Let's go through the answers for each part based on the diagram provided.

Part 1–5: Planes and Segments

1. Plane parallel to $WXT$:

   • Answer: Plane $ZYU$, since they are opposite faces of the prism and therefore parallel.

2. Two segments parallel to $\overline{VU}$:

   • Answer: $\overline{TS}$ and $\overline{ZY}$, as they are in the same direction and on parallel planes.

3. Two segments parallel to $\overline{SW}$:

   • Answer: $\overline{TZ}$ and $\overline{VU}$, as they lie in parallel planes and are oriented in the same direction.

4. Two segments skew to $\overline{XY}$:

   • Answer: $\overline{VT}$ and $\overline{WU}$, as they are neither parallel nor intersecting with $\overline{XY}$.

5. Two segments skew to $\overline{VZ}$:

   • Answer: $\overline{XY}$ and $\overline{SW}$, as they do not lie in the same plane and do not intersect with $\overline{VZ}$.

Part 6: Angle Pair Relationships and Transversals

a) $\angle 4$ and $\angle 10$:

• Relationship: Alternate interior angles
• Transversal: $k$

b) $\angle 8$ and $\angle 11$:

• Relationship: Corresponding angles
• Transversal: $m$

c) $\angle 1$ and $\angle 4$:

• Relationship: Alternate exterior angles
• Transversal: $l$

d) $\angle 2$ and $\angle 12$:

• Relationship: Consecutive interior angles
• Transversal: $k$

e) $\angle 5$ and $\angle 7$:

• Relationship: No relationship (they are not related by any transversal)

f) $\angle 2$ and $\angle 13$:

• Relationship: Alternate exterior angles
• Transversal: $m$

Let me know if you'd like more detailed explanations for any part of these answers, or if you have questions about the concepts used here.

Here are five related questions to deepen your understanding:

1. How do you determine if two segments are skew in three-dimensional space?
2. What makes alternate interior angles equal when two parallel lines are cut by a transversal?
3. Can two segments on parallel planes be skew? Why or why not?
4. What is the significance of identifying corresponding angles in geometry?
5. How can the properties of a prism help in identifying parallel and skew lines?

Tip: To visualize skew lines more effectively, try using 3D models or drawings to see how they do not lie in the same plane and do not intersect.