Points, Lines, and Planes
Define point, line, and plane, the three undefined terms of geometry, and use them to define segments, rays, and angles. Understand collinearity, coplanarity, and how postulates about these objects anchor all geometric reasoning.
A lesson of Geometry, in Foundations of Geometry, about 10 minutes.
By the end of this lesson, you will be able to identify and name points, lines, line segments, rays, and planes; use correct geometric notation; and apply key postulates about points, lines, and planes.
The Big Idea
Geometry is built from three undefined terms — point, line, and plane — that cannot be defined using simpler terms. Everything else in geometry is built on top of them. Undefined doesn't mean vague; it means these are the foundational primitives that everyone agrees on without formal proof.
Undefined Terms
Point: A location with no size, no dimension. Represented by a dot. Named with a capital letter: point $A$.
Line: A straight path extending infinitely in both directions, with no thickness. Named by two points on it: $\overleftrightarrow{AB}$, or by a lowercase letter: line $m$.
Plane: A flat surface extending infinitely in all directions, with no thickness. Named by three non-collinear points: plane $ABC$, or by a capital script letter: plane $\mathcal{P}$.
Defined Terms
Line segment: Part of a line with two endpoints. $\overline{AB}$ (the bar indicates endpoints).
Ray: Part of a line with one endpoint, extending infinitely in one direction. $\overrightarrow{AB}$ (starts at $A$, passes through $B$, continues).
Opposite rays: Two rays with the same endpoint pointing in opposite directions → they form a line.
Key Notation
| Symbol | Meaning |
|---|---|
| $A$ | Point $A$ |
| $\overleftrightarrow{AB}$ | Line through $A$ and $B$ |
| $\overline{AB}$ | Segment from $A$ to $B$ |
| $\overrightarrow{AB}$ | Ray from $A$ through $B$ |
| $AB$ (no symbol) | The length/measure of segment $AB$ |
Collinear and Coplanar
Collinear: Points on the same line.
Non-collinear: Points not all on the same line.
Coplanar: Points and lines in the same plane.
Two points are always collinear — you can always draw a line through any two points. Three points may or may not be collinear.
Postulates (Accepted Without Proof)
Postulate 1 (Ruler Postulate): Every point on a line can be assigned a real number (coordinate). The distance between two points equals the absolute difference of their coordinates.