Points, Lines, Planes & Geometric Notation

Geometry is built on three ideas so basic that they are not defined with simpler words: the point, the line, and the plane. A point is an exact location with no size. A line is a straight path of points that extends forever in both directions. A plane is a flat surface that extends forever in every direction.

Everything else in geometry — segments, rays, angles, triangles — is built from those three. This lesson is about naming them correctly, because $\overline{AB}$ , $\overrightarrow{AB}$ , and $\overleftrightarrow{AB}$ look similar but mean three different things. Reading the little symbol on top is the whole skill.

The three undefined terms

A point is a location. It has no length, width, or thickness — we draw it as a dot only so we can see it, and we name it with a capital letter like $P$ .

A line is a set of points that goes straight and never ends in either direction. Two points are enough to name a line, because exactly one line passes through any two points.

A plane is a flat surface with no edges — think of a tabletop that keeps going forever. We usually name a plane with a single capital letter or with three points that lie on it.

Reading the notation

The symbol drawn over two letters tells you what the figure is. A plain bar, $\overline{AB}$ , is a segment: it starts at $A$ , ends at $B$ , and has a definite length. An arrow pointing one way, $\overrightarrow{AB}$ , is a ray: it starts at endpoint $A$ and passes through $B$ , continuing forever in that one direction. A double arrow, $\overleftrightarrow{AB}$ , is the whole line through $A$ and $B$ , extending forever both ways.

For segments and lines, the letter order does not matter: $\overline{AB}$ and $\overline{BA}$ are the same segment. For rays, the order is everything. A ray is named endpoint first, so $\overrightarrow{AB}$ starts at $A$ , while $\overrightarrow{BA}$ starts at $B$ and heads the opposite way. They are different rays.

The figure stacks all three: a segment that stops at both ends, a ray with one arrowhead, and a line with arrowheads on both ends.

Collinear and coplanar

Points that lie on the same line are collinear. Points that lie on the same plane are coplanar. Any two points are automatically collinear — one line always fits through them — so these words only become a real question with three or more points.

On a diagram, check collinearity by following the drawn line: if the line passes through all the points in question, they are collinear. A point sitting off the line is not collinear with the points on it, even if it is close.

Worked examples

Example 1: write all three notations

Points $C$ and $D$ are given. Write the notation for the segment with endpoints $C$ and $D$ , the ray from $C$ through $D$ , and the line through $C$ and $D$ .

The segment has two endpoints, so use a plain bar

\overline{CD}

The ray starts at

C

, so

C

is written first with a one-way arrow

\overrightarrow{CD}

The line extends both ways, so use a double arrow

\overleftrightarrow{CD}

Answer: $\overline{CD}$ , $\overrightarrow{CD}$ , $\overleftrightarrow{CD}$

Example 2: same letters, different rays

Are $\overrightarrow{AB}$ and $\overrightarrow{BA}$ the same ray?

A ray is named endpoint first, so find each ray's endpoint

\overrightarrow{AB}

has endpoint

A

and heads toward

B

\overrightarrow{BA}

has endpoint

B

and heads toward

A

Different endpoints and opposite directions mean different rays

Answer: No — they are different rays.

Example 3: spotting collinear points

Sprinkler heads $F$ , $G$ , and $H$ sit along one straight pipe. Valve $J$ sits off to the side of the pipe. Which points are collinear?

The straight pipe is a line through the points on it

\overleftrightarrow{FH}

F

G

, and

H

all lie on that line, so they are collinear

J

is off the line, so it is not collinear with them

Answer: $F$ , $G$ , and $H$ are collinear.

Try one yourself

Common questions

Why are point, line, and plane called undefined terms?

Every definition uses simpler words, and these three are as simple as geometry gets — there is nothing more basic to define them with. Instead we describe them and agree on their properties, then use them to define everything else, like segments and angles.

Does the order of the letters matter?

For segments and lines, no: $\overline{AB} = \overline{BA}$ and $\overleftrightarrow{AB} = \overleftrightarrow{BA}$ . For rays, yes: the first letter must be the endpoint, so $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are different rays.

What is the difference between collinear and coplanar?

Collinear points lie on the same line; coplanar points lie on the same plane. Collinear points are always coplanar too, since any line sits inside some plane — but coplanar points are not always collinear.

How is a segment different from a line?

A segment is a piece of a line with two endpoints, so it has a measurable length. A line has no endpoints and no length — it goes on forever, which is why we never talk about how long a line is.

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