In the image, we are given a diagram of a cube with labeled points $A$, $B$, $C$, $D$, and $E$. We are told that:

• $\overline{DE} \parallel \overline{AB}$ (line DE is parallel to line AB).
• $\overline{DE} \perp \overline{EB}$ (line DE is perpendicular to line EB).

The question asks about the relationship between $\overline{AB}$ and $\overline{EB}$, with the possible choices being:

• A) Parallel ($\parallel$),
• B) Perpendicular ($\perp$),
• C) Neither; they are skew lines.

Solution:

• $\overline{AB}$ and $\overline{EB}$ are two edges of the cube, with $\overline{AB}$ being along the base and $\overline{EB}$ going vertically from $E$ to $B$.
• Skew lines are lines that are neither parallel nor intersecting (and hence not perpendicular). In the context of a cube, edges that are not on the same plane and do not meet are skew lines.

Given that $\overline{AB}$ and $\overline{EB}$ are on different planes and do not intersect, the correct answer is C: they are skew lines.

Would you like further clarification on any part of this problem or additional details?

Here are some related questions for further exploration:

1. What is the formal definition of skew lines?
2. How can we identify parallel lines in a 3D shape like a cube?
3. What is the difference between perpendicular and skew lines?
4. How many pairs of skew lines can be identified in a cube?
5. How does the concept of planes help in understanding skew lines in 3D?

Tip: When dealing with 3D geometry, it's important to visualize how the lines are positioned in relation to each other to determine their relationship (parallel, perpendicular, or skew).