Question
Planes M and N intersect at line d. Line a forms a right angle with line d. Line b is diagonal and crosses line d. Line c is on plane N and is diagonal. Line e is on plane M and is slightly diagonal at the top of the plane.
Planes M and N intersect at line d. Line a forms a right angle with line d. Line b is diagonal and crosses line d. Line c is on plane N and is diagonal. Line e is on plane M and is slightly diagonal at the top of the plane.
Which are skew lines? Check all that apply.
a and b
c and b
b and e
e and c
a and e
a and c
Planes M and N intersect at line d. Line a forms a right angle with line d. Line b is diagonal and crosses line d. Line c is on plane N and is diagonal. Line e is on plane M and is slightly diagonal at the top of the plane.
Which are skew lines? Check all that apply.
a and b
c and b
b and e
e and c
a and e
a and c
Answers
GPT-4o mini
To determine which pairs of lines are skew lines, we first need to understand the definitions and relationships of the lines and planes:
- **Skew lines** are lines that do not intersect and are not parallel; they must exist in different planes.
- Lines a and b are related to line d. Line a is perpendicular to line d, and line b crosses line d.
- Line c lies in plane N, which intersects line d.
- Line e lies in plane M, which also intersects line d.
Let's evaluate the pairs of lines:
1. **a and b**: Line a is perpendicular to line d, while line b crosses line d. These lines are not skew since they are both related to the same line d.
2. **c and b**: Line c is in plane N and intersects line d, and line b also intersects line d (it crosses line d). Therefore, c and b are not skew.
3. **b and e**: Line b crosses line d (and is in the plane that intersects with line d), while line e is on plane M and does not intersect line b. Since they are in different planes and do not intersect, b and e are skew lines.
4. **e and c**: Line e is on plane M, and line c is on plane N. Since both lines lie in different planes and do not intersect, e and c are skew lines.
5. **a and e**: Line a is perpendicular to line d and intersects it. Line e exists in plane M but does not intersect line a. Thus, lines a and e are not skew because they are coincident with plane M.
6. **a and c**: Line a is perpendicular to line d, while line c exists in plane N. Since a and c are in different planes, and a is perpendicular to the intersection line d, they cannot intersect; hence, they are skew lines.
From our analysis, the pairs that are skew lines are:
- b and e
- e and c
- a and c
So the final answer is that the skew lines are:
- **b and e**
- **e and c**
- **a and c**
- **Skew lines** are lines that do not intersect and are not parallel; they must exist in different planes.
- Lines a and b are related to line d. Line a is perpendicular to line d, and line b crosses line d.
- Line c lies in plane N, which intersects line d.
- Line e lies in plane M, which also intersects line d.
Let's evaluate the pairs of lines:
1. **a and b**: Line a is perpendicular to line d, while line b crosses line d. These lines are not skew since they are both related to the same line d.
2. **c and b**: Line c is in plane N and intersects line d, and line b also intersects line d (it crosses line d). Therefore, c and b are not skew.
3. **b and e**: Line b crosses line d (and is in the plane that intersects with line d), while line e is on plane M and does not intersect line b. Since they are in different planes and do not intersect, b and e are skew lines.
4. **e and c**: Line e is on plane M, and line c is on plane N. Since both lines lie in different planes and do not intersect, e and c are skew lines.
5. **a and e**: Line a is perpendicular to line d and intersects it. Line e exists in plane M but does not intersect line a. Thus, lines a and e are not skew because they are coincident with plane M.
6. **a and c**: Line a is perpendicular to line d, while line c exists in plane N. Since a and c are in different planes, and a is perpendicular to the intersection line d, they cannot intersect; hence, they are skew lines.
From our analysis, the pairs that are skew lines are:
- b and e
- e and c
- a and c
So the final answer is that the skew lines are:
- **b and e**
- **e and c**
- **a and c**