Lessons 49–50 · 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
50
Perpendicular planes in space
Textbook, Part 2: pp. 132–136
GoalKnow the dihedral angle and its linear angle, the angle between planes, and the criterion and properties of perpendicular planes.
A figure formed by two half-planes with a common boundary line (the edge) is a dihedral angle. Its measure is its linear angle: the angle formed by rays in the two faces drawn perpendicular to the edge from any point of the edge; it is the same wherever on the edge the point is chosen. Two intersecting planes form four dihedral angles; the one not exceeding 90° is the angle between the planes. If it is 90°, the planes are perpendicular (α ⊥ β). Criterion: if one plane passes through a line perpendicular to another plane, the planes are perpendicular. Corollary: a plane perpendicular to the line of intersection of two planes is perpendicular to each of them. The perpendicular dropped from a point of one of two perpendicular planes to the other lies in the first plane.
Worked examples
In a cube planes ABC₁D₁ and ABCD intersect along edge AB. BC ⊥ AB and BC₁ ⊥ AB, so ∠CBC₁ is the linear angle. Since CC₁ = BC, it is 45°.
A regular quadrilateral pyramid has base side 6 and height 3. The angle between a lateral face and the base: the height and the distance from the base centre to the side’s midpoint (r = 3) are equal — a right isosceles triangle, so the angle is 45°.
Class activity
Open a book or a notebook cover partly and hold it at different angles: draw lines perpendicular to the spine in the two covers and measure the linear angle of the dihedral angle with a protractor.
Practice
1
What is the linear angle of a dihedral angle?
The angle between rays drawn in each face perpendicular to the edge from one point of the edge.
2
How many degrees is the angle between two adjacent faces of a cube?
90°
3
A regular quadrilateral pyramid has base side 6 and height √3. Find the angle between a lateral face and the base.
r = 3, tan φ = √3/3, φ = 30°.
4
Why is a plane through a line perpendicular to another plane perpendicular to it?
At a point of the line of intersection take the line in the second plane perpendicular to it; it forms the linear angle with the given perpendicular line; since the perpendicular is perpendicular to every line of the second plane, this angle is 90°.