Points, lines, planes and segments

Three words we don't define

Every definition uses other words, and those words need definitions too. To avoid going around in circles, geometry starts from three undefined terms: point, line and plane. We describe them instead of defining them.

• A point marks a location and has no size. It's named with a capital letter: point \(A\).
• A line is perfectly straight, has no thickness and extends forever in both directions. It's named by two of its points, \(\overleftrightarrow{AB}\), or by a lowercase letter, line \(\ell\).
• A plane is a flat surface with no thickness that extends forever. It's named by three points not on one line, or by a capital script letter.

Points on the same line are collinear. Points in the same plane are coplanar.

The first postulates

A postulate (or axiom) is a statement accepted without proof. Everything else in geometry is built on postulates like these.

Rays and segments

A segment has two endpoints, so it has a length, written \(AB\) (no bar). A ray has only one endpoint and no length. Two rays with a common endpoint that point in exactly opposite directions are opposite rays.

The segment addition postulate

This simple fact turns many geometry questions into algebra.

Congruent segments and midpoints

Segments with the same length are congruent, written \(\overline{AB} \cong \overline{CD}\).

On a number line, the midpoint is the average of the two coordinates. Points at \(-3\) and \(7\) have midpoint \(\frac{-3 + 7}{2} = 2\).

Practice