Fix the mistake: Properties of Geometric Figures ② (Solid Figures and Polyhedra)
Basics of Solid Geometry (Lines and Planes)
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)