Math Core

Every subject needs a starting vocabulary, and geometry's is surprisingly small. Three ideas, point, line and plane, are left undefined on purpose. Every other figure in the course, from a triangle to a sphere, is built out of them.

The three undefined terms

You can't define everything. A definition uses other words, and those words need definitions too, so at some point you have to stop. Geometry stops at three terms. Instead of defining them, it describes them and agrees on how they behave.

A point marks an exact location. It has no size at all. You draw it as a dot and name it with a capital letter, like point $A$ .
A line is perfectly straight, has no thickness and extends forever in both directions. You name it with any two points on it, as $\overleftrightarrow{AB}$ , or with a single lowercase script letter, like line $\ell$ .
A plane is a perfectly flat surface with no thickness that extends forever in every direction. Think of a tabletop that never ends. You name it with a single capital letter, like plane $M$ , or with three points in it that are not on one line, like plane $ABC$ .

Points A, B and C lie on line l. Point D does not.

In the figure, the line can be named $\overleftrightarrow{AB}$ , $\overleftrightarrow{BC}$ , $\overleftrightarrow{CA}$ , $\overleftrightarrow{BA}$ or line $\ell$ . Any two of its points will do, in either order, because a line has no starting end.

Definition

Collinear and coplanar

Points are collinear if they all lie on one line. Points (or lines) are coplanar if they all lie in one plane.

In the figure, $A$ , $B$ and $C$ are collinear. $A$ , $B$ and $D$ are not, since no single line passes through all three. But $A$ , $B$ and $D$ are coplanar: they all lie on the flat page.

Segments and rays

Lines go on forever, but most figures you draw have ends. Two more terms, defined from the undefined ones, handle that.

A segment $\overline{AB}$ is made of the points $A$ and $B$ (its endpoints) and all the points of $\overleftrightarrow{AB}$ between them.
A ray $\overrightarrow{AB}$ starts at the endpoint $A$ and extends forever through $B$ .

A segment stops at both endpoints. A ray starts at R and keeps going through S (it runs off the right edge here). A line keeps going both ways.

For segments, the order of the letters doesn't matter: $\overline{PQ}$ and $\overline{QP}$ are the same segment. For rays it matters a lot. The first letter is always the endpoint, and the second letter only tells you the direction. So $\overrightarrow{RS}$ starts at $R$ and heads toward $S$ , while $\overrightarrow{SR}$ starts at $S$ and heads the other way.

If point $B$ lies between $A$ and $C$ on a line, then $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays. They share an endpoint and together make up the whole line.

Common mistake

Don't treat $\overrightarrow{AB}$ and $\overrightarrow{BA}$ as the same ray. They have different endpoints and point in opposite directions. When you name a ray, say the endpoint first, every time.

How points, lines and planes fit together

A few basic rules, called postulates (or axioms), are accepted without proof. Everything else in geometry is proved from them.

Basic postulates

Through any two points there is exactly one line.
Through any three noncollinear points there is exactly one plane.
If two points lie in a plane, the whole line through them lies in that plane.
If two distinct lines intersect, they intersect in exactly one point.
If two distinct planes intersect, they intersect in exactly one line.

Postulate 2 explains why a three-legged stool never wobbles: the three feet are three noncollinear points, and they always fit on one flat floor. A four-legged chair can rock, because four points don't have to be coplanar.

Postulate 5 is easy to see in any room. Two walls meet along a vertical line in the corner, and a wall meets the floor along a line at the baseboard.

Points A, B and C lie in plane M, so they are coplanar. Point P is above the plane, not in it.

Examples

Worked example: Naming and describing

Use the first figure (line $\ell$ with points $A$ , $B$ , $C$ and point $D$ off the line).

Name the line in two other ways.
Name three collinear points.
Are $B$ , $C$ and $D$ collinear? Are they coplanar?

Solutions.

For example, $\overleftrightarrow{AC}$ and $\overleftrightarrow{CB}$ .
$A$ , $B$ and $C$ .
They are not collinear, because $D$ is not on line $\ell$ . They are coplanar: any three points lie in some plane.

Worked example: Opposite rays

Points $A$ , $B$ and $C$ lie on a line in that order. Which pair of rays are opposite rays: $\overrightarrow{AB}$ and $\overrightarrow{AC}$ , or $\overrightarrow{BA}$ and $\overrightarrow{BC}$ ?

$\overrightarrow{AB}$ and $\overrightarrow{AC}$ both start at $A$ and head the same way (toward $B$ and $C$ ), so they are actually the same ray. $\overrightarrow{BA}$ and $\overrightarrow{BC}$ both start at $B$ and head in opposite directions, so they are the opposite rays.

Worked example: Counting lines

Five points lie in a plane, and no three of them are collinear. How many different lines pass through at least two of the points?

Each line is fixed by a pair of points (Postulate 1), and since no three points are collinear, no line contains more than two of them. So count the pairs. Each of the $5$ points pairs with $4$ others, giving $5 \cdot 4 = 20$ , but that counts every pair twice ( $AB$ and $BA$ ). So there are

$\frac{5 \cdot 4}{2} = 10 \text{ lines.}$

Tip

The same counting works for any number of points with no three collinear: $n$ points give $\dfrac{n(n-1)}{2}$ lines.

Practice

Practice 1

Which is the correct name for the ray that starts at $P$ and passes through $Q$ ?

Practice 2

Two different planes intersect. What is their intersection?

Practice 3

Which of these is always coplanar?

Practice 4

Points $A$ , $B$ and $C$ lie on a line in that order. Which statement is true?

Practice 5

Four points lie in a plane, and no three of them are collinear. How many different lines pass through at least two of them?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Points $A$ , $B$ , $C$ , $D$ and $E$ lie on a line in that order. How many different segments have both endpoints among these five points?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Points $A$ , $B$ , $C$ and $D$ lie on a line in that order. How many different rays have an endpoint at one of these points and pass through another one of them?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Six points lie in a plane, and no three of them are collinear. How many different lines pass through at least two of them?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.