Question
Horizontal plane A and vertical plane B intersect at a line. Line n is vertical on plane B and forms a right angle with horizontal line m on plane A. Line l is diagonal on plane A.
Are the lines in the diagram perpendicular, parallel, skew, or none of these?
l and m:
l and n:
m and n:
Answers
GPT-4o mini
To analyze the relationships between the lines l, m, and n based on the given description:
1. **Lines l and m**:
- Line m is horizontal on plane A.
- Line l is diagonal on plane A.
- Since line l is diagonal, it cannot be parallel to line m, and it does not form a right angle with line m. Therefore, lines l and m are neither parallel nor perpendicular, and they intersect at some angle.
- **Relationship**: None of these (not perpendicular, not parallel).
2. **Lines l and n**:
- Line n is vertical on plane B and line l is diagonal on plane A.
- These lines are not in the same plane (one is in plane A, the other in plane B), and they do not intersect in a way that creates a perpendicular angle.
- Therefore, lines l and n are considered **skew lines** (they are not parallel and do not intersect).
- **Relationship**: Skew.
3. **Lines m and n**:
- Line m is horizontal on plane A.
- Line n is vertical on plane B.
- A horizontal line (m) and a vertical line (n) that meet at a right angle (given that n is vertical on plane B and forms a right angle with the horizontal line) are perpendicular.
- **Relationship**: Perpendicular.
In summary:
- Lines l and m: None of these
- Lines l and n: Skew
- Lines m and n: Perpendicular
1. **Lines l and m**:
- Line m is horizontal on plane A.
- Line l is diagonal on plane A.
- Since line l is diagonal, it cannot be parallel to line m, and it does not form a right angle with line m. Therefore, lines l and m are neither parallel nor perpendicular, and they intersect at some angle.
- **Relationship**: None of these (not perpendicular, not parallel).
2. **Lines l and n**:
- Line n is vertical on plane B and line l is diagonal on plane A.
- These lines are not in the same plane (one is in plane A, the other in plane B), and they do not intersect in a way that creates a perpendicular angle.
- Therefore, lines l and n are considered **skew lines** (they are not parallel and do not intersect).
- **Relationship**: Skew.
3. **Lines m and n**:
- Line m is horizontal on plane A.
- Line n is vertical on plane B.
- A horizontal line (m) and a vertical line (n) that meet at a right angle (given that n is vertical on plane B and forms a right angle with the horizontal line) are perpendicular.
- **Relationship**: Perpendicular.
In summary:
- Lines l and m: None of these
- Lines l and n: Skew
- Lines m and n: Perpendicular