We will accurately understand the positional relationships between lines and planes in space.
Positional Relationships Between Lines
There are three types of relationships between lines in space:
①Intersecting（intersecting at one point）
②Parallel（do not intersect, lie in the same plane）
③Skew（do not intersect, do not lie in the same plane）
Example ① (Relationships between lines in a cube)
In cube ABCD−EFGH, state the positional relationship between edge AB and edge DH.
AB is in the x-direction, and DH is in the y-direction (different planes) → they do not intersect and do not lie in the same plane
Answer:Skew
In cube ABCD−EFGH, state the positional relationship between edge AB and edge DH.
AB is in the x-direction, and DH is in the y-direction (different planes) → they do not intersect and do not lie in the same plane
Answer:Skew
Positional Relationships Between a Line and a Plane
①The line lies in the plane
②The line intersects the plane at one point
③The line is parallel to the plane (does not intersect)
Line ⊥ plane: the line is perpendicular to every line in the plane
Positional Relationships Between Planes
①The two planes coincide ②The two planes intersect (along their line of intersection) ③The two planes are parallel
Example ② (Line ⊥ plane)
In △ABC, given AB⊥BC and AB⊥BD（D is a point outside the plane）, prove that AB ⊥ plane BCD.
AB is perpendicular to both lines BC and BD, which intersect in plane BCD, so AB ⊥ plane BCD ■
In △ABC, given AB⊥BC and AB⊥BD（D is a point outside the plane）, prove that AB ⊥ plane BCD.
AB is perpendicular to both lines BC and BD, which intersect in plane BCD, so AB ⊥ plane BCD ■
Key points
- Skew lines: they neither intersect nor are parallel (unique to space)
- Line ⊥ plane ⟺ perpendicular to two intersecting lines in the plane
- Angle between two planes: measure it in a cross-section perpendicular to their line of intersection
Practice Problems
- In cube ABCD−EFGH, name all edges skew to edge AB.
- When line l is perpendicular to plane α and line m lies in plane α, state the relationship between l and m.
- When planes α and β have line of intersection l, state the relationship between l and α.
Answers and Explanations
- Edges that do not intersect AB and are not parallel to it: CG、DH、EF（EH is parallel; BC,AD,AE,BF intersect） → the three edges CG, DH, and EF
- l ⊥ α and m lies in α → l ⊥ m (perpendicular)
- Line of intersection l is a line in plane α (it is contained in α)