Planes

A plane is a fundamental concept in geometry characterized by two dimensions: length and width.
an example of a plane

A plane extends infinitely in all directions and has no thickness or boundaries.

It is usually represented by a letter from the Greek alphabet (e.g., α, β, γ, δ, etc.).

a line and a point on a plane

When a line or point lies on a plane, it is said to belong to the plane.

For example, line r belongs to plane alpha.

How is a Plane Defined?

A plane is defined by three non-collinear points (A, B, C) because exactly one plane α passes through any three non-collinear points.

an example of a plane determined by three points

Alternatively, a plane can also be defined by a line and a point not on that line.

For a line r and a point P not on the line, there is exactly one plane α that contains both.

a point P and a line

Finally, a plane can also be defined by two intersecting lines.

Through two intersecting lines r and s, there is exactly one plane α.

two intersecting lines

A half-plane is a portion of a plane divided by a line.
the half-plane

In other words, when a line is drawn on a plane, it divides the plane into two distinct parts, each of which is a half-plane.

Depending on their position in space, two distinct planes are either intersecting or parallel.

Intersecting Planes
Two planes intersect if they share exactly one line.
Parallel Planes
Two planes are parallel if they have no points in common (non-coincident parallel planes) or all points in common (coincident planes).

The distance between two parallel planes is the length of the segment perpendicular to both planes that connects a point on each plane.

The relationship of parallelism between planes has reflexive, symmetric, and transitive properties.

Reflexive Property
Every plane is parallel to itself $$ \alpha || \alpha $$
Symmetric Property
If plane α is parallel to plane β, then plane β is parallel to plane α $$ \alpha || \beta \Leftrightarrow \beta || \alpha $$
Transitive Property
If plane α is parallel to plane β and plane β is parallel to plane γ, then plane α is parallel to plane γ $$ \alpha || \beta \ , \ \beta || \gamma \Rightarrow \alpha || \gamma $$

A line in the plane can be:

Coplanar Line (or lying on the plane)
The line lies on the plane. All points on the line are also on the plane r⋂α=r.
Intersecting Line
The line and the plane share exactly one point r⋂α={P}. An intersecting line can be perpendicular or oblique to the plane.
- Line Perpendicular to a Plane
  If the line is perpendicular to every line in the plane that passes through the point of intersection (called the foot of the perpendicular).
- Oblique Line
  If the line is not perpendicular to the plane.
Line Parallel to the Plane
The line has no points in common with the plane r⋂α=Ø.

And so on.