## Wednesday, June 17, 2015

### Unizor - Geometry3D - Lines and Planes - Perpendicular Planes

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Definitions

1. A dihedral angle is called the right dihedral angle if the corresponding linear angle is the right angle (900 = π/2 radian).

2. Two half-planes with a common edge are called perpendicular to each other if a dihedral angle they form is the right dihedral angle.

3. Two intersecting planes are called perpendicular to each other if any one of four pairs of their half-planes with common edge along the line of intersection are perpendicular to each other.

There are four dihedral angles formed by four pairs of half-planes of two intersecting planes. Obviously, it's sufficient to have only one of these four to form the right dihedral angle for all four to be the same. This follows from the corresponding properties of linear angles - if one of two supplemental angles is the right angle, the other is the right angle as well.

Theorem 1

Given a plane σ, a point A on it and a line p perpendicular to plane σ at point A.

Prove that any plane that contains line p is perpendicular to plane σ.

The above theorem stated that a plane that contains a line perpendicular to another plane is itself perpendicular to that plane.

So, from a perpendicular line we went to a perpendicular plane.

We can go the other way around, from a perpendicular plane go to a perpendicular line.

Theorem 2

Given two planes σ and τ perpendicular to each other (that is, the linear angle of any dihedral angle they form is the right angle.)

Then there exists a line s contained in plane σ that is also perpendicular to plane τ.

Theorem 3

Given two planes σ and τ perpendicular to each other (that is, the linear angle of any dihedral angle they form is the right angle.)

Choose any point A on plane σ outside of the line of intersection d of these two planes and drop a perpendicular from this point A onto plane τ. Then this perpendicular is completely contained in plane σ.

Theorem 4

Given two planes σ and τ intersecting along line p (σ∩τ = p).

Another given plane γ is perpendicular to both planes σ and τ (σ⊥γ and τ⊥γ).

Then the line of intersection p of planes σ and τ is perpendicular to plane γ.

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