The Timpson Lyceum · Geometry (2024)

Postulates and Definitions Concerning Lines – Lecture 4 (Geometry)

Overview

This lecture launches the formal study of geometry as an axiomatic system and introduces the first body of postulates and definitions concerning lines, betweenness, and angles. Geometry is taught precisely because it is one of the simplest axiomatic systems to reason about: its pictorial nature lets us build intuition before we prove. You will see why every systematic field rests on undefined terms and basic assumptions, meet the undefined terms point, line, plane, and between, and work through the first group of postulates and theorems. By the end you can prove small results about lines (e.g. that two lines meet in at most one point) and you have the definitions needed to talk about sides of a line, segments, rays, angles, and their interiors.

Key Concepts

The four components of an axiomatic system

Undefined terms — basic notions left undefined to avoid circular or infinitely regressive definitions (like trying to learn French from a French-only dictionary). They are not meaningless; the postulates give them meaning.
Postulates (axioms) — the starting assumptions you are allowed to reason from. "Postulate" and "axiom" mean the same thing.
Definitions — new words built from existing terms. Convenient (they compress meaning) but never strictly necessary.
Theorems — the logical consequences of the postulates; deriving them is the heart of mathematics.

Undefined terms in this geometry

Point, line, plane — left undefined. A line is thought of as a set of points; a plane is also thought of as a set of points.
Between — a three-term predicate on points. Writing $A\text{-}B\text{-}C$ means " $B$ is between $A$ and $C$ ." The notation is reserved for exactly three points; never string more letters together to mean chained betweenness.

Postulate group 1 (lines, points, planes)

Postulate I.1 — In the plane there exist three non-collinear points (three points not all on one line).
Postulate I.2 — For any two points there is exactly one line containing them.
Postulate I.3 — Every line contains at least two points.
Guiding principle: do not read more into the figure than what the postulates give you. Pictures aid intuition but are never part of a proof.

Postulate group 2 (betweenness)

Postulate II.1 — If $A\text{-}B\text{-}C$ , then $A,B,C$ are three distinct points lying on a single line.
Postulate II.2 — If $A,B,C$ are points on a line, exactly one of $A\text{-}B\text{-}C$ , $B\text{-}A\text{-}C$ , or $A\text{-}C\text{-}B$ holds (one of the three is the "middle" point). Also $A\text{-}B\text{-}C$ is equivalent to $C\text{-}B\text{-}A$ — order of the endpoints does not matter.
Postulate II.3 — Given four points on a line, they can be relabeled $A_1,A_2,A_3,A_4$ so that the only betweenness relations are $A_1\text{-}A_2\text{-}A_3$ , $A_1\text{-}A_2\text{-}A_4$ , $A_2\text{-}A_3\text{-}A_4$ , and $A_1\text{-}A_3\text{-}A_4$ .
Postulate II.4 (line-builder) — For any two points $A,B$ there exists a point $C$ with $A\text{-}B\text{-}C$ and a point $D$ with $A\text{-}D\text{-}B$ . Applied repeatedly, this guarantees infinitely many points on the line, in both directions.
Postulate II.5 (line separates plane) — Each line separates the points of the plane not on it into two sets; two points are on the same side iff they lie in the same set.

Definitions concerning sides, segments, and rays

Same/opposite sides of a line (Def. 3.1) — For $A,B$ not on line $L$ : $A$ and $B$ are on opposite sides of $L$ iff there is a point $C \in L$ with $A\text{-}C\text{-}B$ ; otherwise they are on the same side.
Segment (Def. 3.2) — $\overline{AB} = \{P : P = A \text{ or } P = B \text{ or } A\text{-}P\text{-}B\}$ : the endpoints together with every point between them.
Sides of a point on a line — the two sets produced by a point $O$ on $L$ (proved possible in Theorem 3.3).
Ray — point $O$ together with all points on one side of $O$ along a line through $O$ (a "half-line" with its endpoint).

Definitions concerning angles

Angle — the set of all points on two rays sharing a common endpoint $O$ (the vertex); named $\angle AOB$ for points $A,B$ on the two rays. The rays are the sides.
Straight angle — when the two rays lie on one line.
Vertical angles — the pairs of opposite angles formed by two intersecting lines (not merely angles of equal size).
Adjacent angles — two angles sharing a vertex and a common ray, with their other rays on opposite sides of the line carrying the common ray.
Supplementary angles — adjacent angles whose non-common sides lie on a single line.
Interior of an angle — for an angle with rays $R_1,R_2$ on lines $L_1,L_2$ : all points on the same side of $L_1$ as $R_2$ and the same side of $L_2$ as $R_1$ . Points neither interior nor on the angle are exterior — a point may be on the angle itself, so "not interior" does not force "exterior."

Worked Example

Theorem 3.1 — Two distinct lines intersect in at most one point.

Strategy: assume the opposite and derive a contradiction with Postulate I.2 (exactly one line through two points).

\begin{aligned} &\textbf{1.}\ \text{Let } l, m \text{ be two distinct lines.} \\ &\textbf{2.}\ \text{Assume by way of contradiction that } l \text{ and } m \text{ meet in two points } A, B. \\ &\textbf{3.}\ \text{Then both } l \text{ and } m \text{ contain the points } A \text{ and } B. \\ &\textbf{4.}\ \text{By Postulate I.2 there is } \textit{exactly one} \text{ line containing } A \text{ and } B \;\Rightarrow\!\Leftarrow \\ &\textbf{5.}\ \therefore\ l \text{ and } m \text{ intersect in at most one point.} \quad \blacksquare \end{aligned}

Theorem 3.2 — There are at least three lines in a plane (proof sketch).

By Postulate I.1 there exist three non-collinear points $A, B, C$ . By Postulate I.2 each pair determines a line, giving $\overline{AB}$ , $\overline{AC}$ , $\overline{BC}$ . If any two of these were the same line, all three points would lie on one line, contradicting their non-collinearity (Postulate I.1). Hence the three lines are distinct. $\blacksquare$

Summary

Every systematic body of knowledge is (or presupposes) an axiomatic system: undefined terms, postulates, definitions, and theorems.
The undefined terms here are point, line, plane, and between; lines and planes are treated as sets of points, and betweenness $A\text{-}B\text{-}C$ is a three-point predicate.
Group 1 postulates: three non-collinear points exist; two points determine exactly one line; every line has at least two points.
Group 2 postulates make betweenness rigorous and, via the line-builder postulate, force infinitely many points on every line; the separation postulate splits the plane into two sides of any line.
Key definitions build upward: same/opposite sides → segment → ray → angle, with the interior of an angle defined by the sides of its two carrier lines.
Proof discipline: draw the figure ugly to avoid over-reading it, prove until a claim feels obvious, and make the written proof stand on the postulates alone — the picture is intuition, never evidence.