Lessons 22–23 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
22
Lines and planes in space
Textbook, Part 1: pp. 126–130
GoalKnow the possible positions of lines and planes in space, the axioms of stereometry and the ways a plane is determined.
Two lines in space can be placed in three ways: they intersect (one common point), they are parallel (lie in one plane and have no common point) or they are skew (do not lie in any single plane). A line and a plane also have three positions: the line lies in the plane, meets it at one point, or has no common point (parallel). Two planes either intersect along a common line or have no common point and are parallel. Three rules are accepted without proof: exactly one plane passes through three points not on one line; if two points of a line lie in a plane, the whole line lies in it; if two planes have a common point, they also have a common line through that point. From these follow the ways to determine a plane: three points, a line and a point not on it, two intersecting lines (two parallel lines work as well). No common plane can be drawn through two skew lines.
Worked examples
The lines through edges AB and CC₁ of a cube: they lie in no common plane and do not meet, so they are skew. Edges AB and A₁B₁ lie in one face and do not meet, so they are parallel.
A four-legged table sometimes wobbles, a three-legged table never does: the leg ends are three points not on one line, so exactly one plane passes through them and the table rests on that plane.
Class activity
In pairs model with two pencils and a sheet of paper: place the pencils intersecting, parallel and skew, and each time test whether the sheet (a plane) can be laid through both.
Practice
1
How can two lines in space be placed?
Intersecting, parallel or skew.
2
Through how many points not on one line does exactly one plane pass?
3
3
How are the lines of edges AB and C₁D₁ of a cube placed (ABCD is the bottom, A₁B₁C₁D₁ the top face)?
Parallel (AB ∥ A₁B₁ ∥ C₁D₁).
4
Why can no plane be drawn through two skew lines?
By definition skew lines lie in no single plane; if a common plane existed they would be intersecting or parallel.