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

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Definition 1
A line a is called perpendicular to a plane μ if it intersects this plane at some point M and any line within this plane μ that passes through a point of intersection M is perpendicular to line a.

Definition 2
A non-perpendicular line intersecting a plane is called slant or oblique.

Definition 3
A point of intersection of a line perpendicular to a plane with this plane is called a base of a perpendicular.

Construction 1
Given a line a and a point A on it.
Construct a plane passing through point A perpendicular to line a.

Mini-Theorem 1
There is one and only one plane perpendicular to a given line at a given point.

Construction 2
Given a plane β and a point A on it.
Construct a line a perpendicular to plane β intersecting it at point A.

Mini Theorem 2
There is one and only one line perpendicular to a given plane at a given point.

Theorem
If line a intersects plane μ at some point M (μ∩a=M) and is perpendicular to two different lines, b and c, lying within this plane μ and passing through a point of intersection M (b∈μ; c∈μ; b∩a=M; c∩a=M; b⊥a; c⊥a), then it is perpendicular to any other line n on the plane μ that passes through point M (n∈μ; n∩a=M ⇒ a⊥n) and, consequently, is perpendicular to an entire plane μ (a⊥μ).