Answer :
Answer:If \( \mathbb{R}^3 \) refers to the three-dimensional Euclidean space, the relationships between lines can be described as follows:
1. **Two lines that do not intersect**:
- They can either be parallel or skew.
2. **Two lines lying in the same plane**:
- They can either intersect or be parallel.
3. **Two lines not lying in the same plane**:
- They are skew.
4. **Two lines that are parallel**:
- They lie in the same plane and do not intersect.
5. **Two lines that are perpendicular**:
- They intersect and form a right angle.
6. **Two lines that are skew**:
- They do not intersect and do not lie in the same plane.
Based on this, the correctness of each statement can be assessed:
- A. and do not intersect: This can be true if the lines are either parallel or skew.
- B. and lie in the same plane: This can be true if the lines either intersect or are parallel.
- C. and do not lie in the same plane: This is true if the lines are skew.
- D. and are parallel: This implies the lines do not intersect and lie in the same plane.
- E. and are perpendicular: This implies the lines intersect at a right angle.
- F. and are skew: This implies the lines do not intersect and do not lie in the same plane.
Therefore:
- Statements A, B, C, and D can be true in certain contexts.
- Statement E is true if the lines intersect at a right angle.
- Statement F is true if the lines do not intersect and do not lie in the same plane.
Step-by-step explanation:
