https://file.aiquickdraw.com/mathbot/file/17265627178488xvgb6yp.jpg

Let's break down and answer the questions based on the cube shown in the image.

### Given Information
- A cube is placed on a horizontal table.
- The shaded face of the cube is labeled as \( AHDC \), which lies vertically.

### Questions and Solutions:

**(a) How many faces are parallel to the shaded face?**
- The shaded face is \( AHDC \). A face parallel to this would lie opposite, so the face \( BGFH \) is parallel to \( AHDC \).
- **Answer:** 1 face (the opposite face \( BGFH \)).

**(b) How many faces are perpendicular to the shaded face?**
- Faces perpendicular to \( AHDC \) are those that meet it at a right angle. These would be the top face \( ABCD \), the bottom face \( EFGH \), and the two side faces \( ABFE \) and \( EDCB \).
- **Answer:** 4 faces (the top, bottom, and two side faces).

**(c) Name three pairs of parallel edges.**
- Parallel edges in a cube are those that do not intersect and are on the same plane or aligned in the same direction. Examples:
  1. \( AH \parallel DC \)
  2. \( AD \parallel HC \)
  3. \( AB \parallel FE \)
- **Answer:** \( AH \parallel DC \), \( AD \parallel HC \), \( AB \parallel FE \).

**(d) Name three pairs of perpendicular edges.**
- Perpendicular edges are those that meet at a 90° angle. Examples:
  1. \( AD \perp DC \)
  2. \( AH \perp AB \)
  3. \( HC \perp CD \)
- **Answer:** \( AD \perp DC \), \( AH \perp AB \), \( HC \perp CD \).

Would you like further details or explanations on any of the points?

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### Follow-up Questions:
1. What are the properties of a cube that make faces parallel or perpendicular to each other?
2. How would you find the length of the edges if one face area is known?
3. How many edges in total does a cube have?
4. Can any two diagonal edges be parallel in a cube?
5. How many diagonals does a cube have?

### Tip:
To easily identify parallel edges, look for edges that are aligned in the same direction but on different faces.

This cube is placed on a horizontal table. (a) How many faces are parallel to the shaded face? (b) How many faces are perpendicular to the shaded face? (c) Name three pairs of parallel edges. (d) Name three pairs of perpendicular edges.

Learn how to analyze the parallel and perpendicular faces and edges of a cube placed on a horizontal table. This geometry problem explores key concepts such as parallelism and perpendicularity, making it suitable for students in Grades 5-7.

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