Points, Lines & Planes

Undefined terms, Euclidean postulates, and the betweenness that adds segments up.

The idea

Points, lines, and planes are geometry's three basic objects, the undefined terms. A point marks a position and has no size. A line is straight, has no thickness, and extends without end in both directions. A plane is a flat surface extending without end in every direction. We do not define them — any definition would use words that themselves need defining — and instead state how they behave. Each statement accepted without proof is a postulate.

Three postulates do most of the work. Two distinct points lie on exactly one line. Through a point not on a given line there is exactly one line parallel to it — the Parallel Postulate. And if $B$ lies between $A$ and $C$, then $AB + BC = AC$ — the Segment Addition Postulate: a segment is the sum of its two pieces.

These postulates describe the Euclidean plane, the flat geometry of a sheet of paper; on a curved surface such as a sphere the first two fail. Every other object — segment, angle, triangle — is defined from the undefined terms, and every later fact is proved from the postulates.

Ways to work on it

• Walkthrough. Undefined terms, the Euclidean postulates, segment addition, and midpoints.
• Practice. Find the missing length with the Segment Addition Postulate.
• Hardest. Midpoint with algebra: solve for x, then the full length.

Not sure where to start? Take the ten-question placement test.