The Timpson Lyceum · Geometry (2024)

Postulates and Definitions Concerning Congruence – Lecture 5 (Geometry)

Overview

This lecture closes out the betweenness material of the previous chapter and then introduces congruence, the central undefined term for comparing geometric figures. After recapping the four ingredients of an axiomatic system (undefined terms, definitions, postulates, theorems) and the major proof that a segment joining an interior point to an exterior point must cross an angle, we adopt congruence as a new undefined term and lay down the three postulate groups governing it: segment construction, congruence as an equivalence relation, and segment addition/subtraction. We then build the order relations greater than and less than for segments using only congruence and betweenness, and prove the trichotomy law together with three comparison theorems. The recurring lesson is that "same size" (congruence) is the sameness relation geometry cares about, and that inequalities for segments must always be unpacked into the precise betweenness statements they encode.

Key Concepts

The four parts of an axiomatic system

Undefined terms — the starting words taken without definition. So far: point, line, plane, between.
Definitions — every new term, built only from undefined terms and previously defined terms.
Postulates (axioms) — statements assumed true; they give the undefined terms their meaning by fixing how they behave.
Theorems — everything else that must hold if the postulates hold.

Recap — interior of an angle and the key prior results

A point $P$ is in the interior of $\angle O$ (sides $R_1$ on line $L_1$ , $R_2$ on line $L_2$ ) if $P$ is on the same side of $L_1$ as $R_2$ , on the same side of $L_2$ as $R_1$ , and not on either ray. The interior is the intersection of those two half-planes.
Two lines meet in at most one point, so a segment cannot cross a line twice.
If $P$ and $Q$ both lie in the interior of an angle, the entire segment $\overline{PQ}$ lies in the interior.
Any ray from the vertex other than $R_1, R_2$ lies entirely in the interior or entirely in the exterior.
Interior of a triangle: the set of points interior to all three of its angles (equivalently, to any two of them). The exterior is every point neither on nor interior to the triangle.

Congruence (new undefined term)

Intuitively, two segments are congruent ( $\cong$ ) when they have the same length. Congruence is written with the squiggle-over-equals symbol and is not equality: $=$ means identical (the same set of points), while $\cong$ allows two distinct segments of the same size.

Postulate group on congruence

Segment construction. Given segment $\overline{AB}$ and a point $A'$ on a line $L'$ , there exist exactly two points (one on each side of $A'$ ) such that the new segment from $A'$ is congruent to $\overline{AB}$ . Restricting to one chosen side gives a unique such point.
Congruence is an equivalence relation, i.e. it satisfies:
- Reflexive: $\overline{AB} \cong \overline{AB}$ .
- Symmetric: if $\overline{AB} \cong \overline{A'B'}$ then $\overline{A'B'} \cong \overline{AB}$ .
- Transitive: if $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$ then $\overline{AB} \cong \overline{EF}$ . An equivalence relation always asserts that two things are the same in some sense; congruence is geometry's chosen sameness relation (like cardinality for sets or homeomorphism for topologies).
Segment addition / subtraction. If $B$ $B$ is between $A$ $A$ and $C$ $C$ , and $B'$ $B^{'}$ is between $A'$ $A^{'}$ and $C'$ $C^{'}$ , then:
- Addition ("equals added to equals are equal"): if $\overline{AB} \cong \overline{A'B'}$ and $\overline{BC} \cong \overline{B'C'}$ , then $\overline{AC} \cong \overline{A'C'}$ .
- Subtraction ("equals subtracted from equals are equal"): if $\overline{AB} \cong \overline{A'B'}$ and $\overline{AC} \cong \overline{A'C'}$ , then $\overline{BC} \cong \overline{B'C'}$ . The subtraction statement is provable from the rest; it is kept as a postulate only for convenience.

Defining segment inequality (the hard definition)

Let $\overline{AB}$ $\overline{A B}$ and $\overline{A'B'}$ $\overline{A^{'} B^{'}}$ be segments on line $L'$ $L^{'}$ , and let $B''$ $B^{''}$ be the point on the same side of $A'$ as $B'$ with $\overline{AB} \cong \overline{A'B''}$ $\overline{A B} ≅ \overline{A^{'} B^{''}}$ . Then:
- $\overline{AB} > \overline{A'B'}$ means $B'$ is between $A'$ and $B''$ (the copy of $\overline{AB}$ overshoots $\overline{A'B'}$ ).
- $\overline{AB} < \overline{A'B'}$ means $B''$ is between $A'$ and $B'$ (the copy falls short).
$\geq$ and $\leq$ are defined as the corresponding strict inequality or congruence.
Inequality is defined purely through congruence and betweenness. The statements $\overline{AB} > \overline{A'B'}$ and $\overline{A'B'} < \overline{AB}$ are logically equivalent but say different things (they encode different betweenness relations), so each must be unpacked carefully — never flipped by reflex.

Comparison theorems

Trichotomy. For any two segments, exactly one holds: $\overline{AB} > \overline{A'B'}$ , $\overline{AB} \cong \overline{A'B'}$ , or $\overline{AB} < \overline{A'B'}$ .
$\overline{AB} > \overline{A'B'}$ if and only if $\overline{A'B'} < \overline{AB}$ .
If $\overline{AC} \cong \overline{A'C'}$ , $B'$ is between $A'$ and $C'$ , and $B$ is the point on $C$ 's side of $A$ with $\overline{AB} \cong \overline{A'B'}$ , then $B$ is between $A$ and $C$ .
If $\overline{AB} > \overline{CD}$ and $\overline{CD} \cong \overline{EF}$ , then $\overline{AB} > \overline{EF}$ .

Worked Example

Theorem. If $\overline{AB} > \overline{CD}$ and $\overline{CD} \cong \overline{EF}$ , then $\overline{AB} > \overline{EF}$ .

The strategy is to translate the inequality into a betweenness fact on a single line, swap the congruent segment in, and translate back.

\begin{aligned} \overline{AB} > \overline{CD} \;&\Longrightarrow\; \overline{CD} < \overline{AB} \\ &\Longrightarrow\; \exists\, B' \text{ with } B' \text{ between } A \text{ and } B \text{ and } \overline{AB'} \cong \overline{CD} \\ \overline{EF} \cong \overline{CD} \cong \overline{AB'} \;&\Longrightarrow\; \overline{EF} \cong \overline{AB'} \quad (\text{transitivity}) \\ \overline{EF} \cong \overline{AB'} \ \text{and}\ B' \text{ between } A, B \;&\Longrightarrow\; \overline{EF} < \overline{AB} \\ &\Longrightarrow\; \overline{AB} > \overline{EF}. \qquad \blacksquare \end{aligned}

The whole proof hinges on reading $\overline{CD} < \overline{AB}$ as the existence of a point $B'$ between $A$ and $B$ with $\overline{AB'} \cong \overline{CD}$ — once $\overline{EF}$ is shown congruent to that same $\overline{AB'}$ , the definition of $<$ delivers the conclusion directly.

Summary

An axiomatic system has four parts: undefined terms, definitions, postulates, and theorems — postulates supply the meaning of the undefined terms.
Congruence is a new undefined term meaning "same size"; it is written $\cong$ and is distinct from $=$ (identity), allowing two different segments to be congruent.
Three postulate groups govern congruence: segment construction (a unique copy on a chosen side), congruence as an equivalence relation (reflexive, symmetric, transitive), and segment addition/subtraction.
Segment inequality is defined entirely from congruence and betweenness: lay a congruent copy from a common endpoint on the matching side and check whether the far endpoint overshoots ( $>$ ) or falls short ( $<$ ).
The trichotomy law guarantees exactly one of $>$ , $\cong$ , $<$ for any two segments, and $\overline{AB} > \overline{A'B'} \iff \overline{A'B'} < \overline{AB}$ — equivalent but not identical statements.
To use any segment inequality, always unpack it into the precise congruence and betweenness facts it stands for rather than manipulating the symbols by analogy with numbers.