https://file.aiquickdraw.com/mathbot/file/17283246821049niymqwu.jpg

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).

When DE is parallel to AB and DE is perpendicular to EB, what is the relationship between AB and EB?

Explore the relationship between lines AB and EB in a 3D cube, where DE is parallel to AB and DE is perpendicular to EB. Understand the concept of skew lines and how it applies to this geometry problem. Suitable for students in Grades 9-12.

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