Parallelism of a line and a plane, and of two planes
Lessons 43–44 · 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
43
Mutual position of a line and a plane in space
Textbook, Part 2: pp. 106–107
GoalKnow the positions of a line and a plane, the criterion and property of a line parallel to a plane and use them in problems.
New words
parallelism of a line and a plane · to‘g‘ri chiziq va tekislik parallelligicriterion · alomatline of intersection · kesishish chizig‘isimilar triangles · o‘xshash uchburchaklar
Explanation
A line and a plane can be placed in three ways: the line lies in the plane, meets it at one point, or has no common point with it. If there is no common point, the line is parallel to the plane. Criterion: a line outside a plane that is parallel to some line of the plane is parallel to the plane itself. Property: if a plane passes through a line parallel to a second plane, the line of intersection of the planes is parallel to that line. So, for example, if a plane parallel to side BC of a triangle meets AB and AC at B₁, C₁, then B₁C₁ ∥ BC and, by similar triangles, B₁C₁ : BC = AB₁ : AB. A line parallel to a plane is not parallel to every line of the plane: some of them are skew to it, and only those in its direction are parallel.
Worked examples
In triangle ABC a plane parallel to BC meets AB and AC at B₁ and C₁; AB₁ : B₁B = 3 : 2 and BC = 20. Then AB₁ : AB = 3 : 5, so B₁C₁ = 20 · 3/5 = 12.
In a cube the line A₁B₁ is parallel to plane ABCD: A₁B₁ ∥ AB, AB lies in ABCD and A₁B₁ does not — the criterion applies.
Class activity
Build a model from a pencil and a tabletop: hold the pencil above the table parallel to a line on it and show that it does not meet the table; then turn it to a position skew to a line on the table.
Practice
1
State the criterion for a line to be parallel to a plane.
A line outside a plane that is parallel to a line in the plane is parallel to the plane.
2
In triangle ABC a plane parallel to BC meets AB and AC at B₁, C₁, with AB₁ : AB = 2 : 5 and BC = 15. Find B₁C₁.
6
3
ABCD is a square and M is a point not in its plane; K and L are the midpoints of MA and MB. Is line KL parallel to plane ABCD? Justify.
Yes: KL is a midline of triangle MAB, so KL ∥ AB; AB lies in the plane and KL does not, so by the criterion KL ∥ (ABCD).
4
Why is a line parallel to a plane not parallel to all lines in that plane?
The plane also contains lines skew to it: they lie in no common plane although they share no point.