# Planes

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

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.).

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.

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.

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

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

## The Half-Plane

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

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.

## Intersecting and Parallel Planes

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 $$ \alpha || \beta \ , \ \beta || \gamma \Rightarrow \alpha || \gamma $$

## The Position of a Line in the Plane

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.