Four kinds of sentence
In the previous guide you met point, line, and plane — the words Euclid refused to define, because you have to start somewhere. Now we widen the lens. Every sentence in geometry is one of exactly four kinds: an undefined term, a definition, a postulate, or a theorem. Knowing which kind a sentence is tells you whether you may simply use it, or whether you owe the world a proof. That single habit — always asking 'what kind of statement is this?' — is most of what separates someone who does geometry from someone who only looks at pictures.
Think of it like a board game fresh out of the box. The undefined terms are the physical pieces — you point at them and say 'this is a pawn' without explaining what a pawn 'really is.' The definitions name the configurations you'll talk about a lot. The postulates are the printed rules you agree to before anyone moves. And the theorems are everything you can prove must follow once those rules are accepted. Crucially, you never prove the rules; you prove with them.
Undefined terms and definitions
Why allow words with no definition at all? Because every definition must be built from simpler words, and if you never stop you spiral forever — 'a point is a location,' but then what is a location? Euclid's honest answer was to pick a tiny handful of undefined terms (point, line, plane) and refuse to define them, fixing their behavior only through the postulates. This is not laziness; it is the price of honesty. You cannot define everything, so you choose your foundation openly instead of pretending it isn't there.
A good definition does two jobs at once: it must include everything it should and exclude everything it shouldn't, using only words already available. 'A midpoint of a segment is the point that divides it into two equal parts' works because point, segment, and 'equal' are already on the table. Definitions are reversible by design — if M is the midpoint then the parts are equal, and if the parts are equal then M is the midpoint. That two-way street is exactly what lets a definition act as a biconditional, a tool you'll lean on in every proof.
Postulates: the rules you agree to
A postulate (or axiom) is a statement accepted without proof so that reasoning can begin. Euclid opened the Elements with five of them. The first four are mild and intuitive: a line can be drawn between any two points; a segment extends to a line; a circle can be drawn with any center and radius; all right angles are equal. Alongside these he listed common notions — even more general truths like 'things equal to the same thing are equal to each other.' A common notion is a postulate so basic it applies far beyond geometry, and you will use it constantly when you write 'AB = CD, and CD = EF, therefore AB = EF.'
The famous fifth — the parallel postulate — is different, and worth meeting honestly even now. It says, roughly, that through a point not on a given line there is exactly one parallel. For two thousand years mathematicians felt it was too complicated to be a mere rule and tried to prove it from the other four. They all failed — not from lack of cleverness, but because it genuinely cannot be derived from them. That failure was eventually one of the great liberations in mathematics: deny it and you get a different, perfectly consistent geometry. We are nowhere near that yet, but it is worth knowing from the start that postulates are choices, not commandments.
Theorems: the earned statements
A theorem is a statement you are not allowed to assume — you must earn it by a chain of reasoning that starts from undefined terms, definitions, postulates, and any theorems already proved. The line between a postulate and a theorem is the whole point of the distinction between a definition and a theorem: a definition is a naming convention you may invoke freely, while a theorem is a claim that demands a proof before you trust it. 'The angles of a triangle add to 180 degrees' is not obvious from the box of rules — it is a theorem, and a beautiful one you'll prove later in this rung.
Here is the structure all together. Read it top to bottom: each layer is allowed to use only the layers above it, never below. This is what people mean by the axiomatic method — certainty manufactured in a strict order, so that a single proof, once correct, stays correct forever.
UNDEFINED TERMS point, line, plane (used, never defined)
DEFINITIONS midpoint, right angle, ... (named from earlier words)
POSTULATES the 5 postulates, common (accepted, never proved)
notions
THEOREMS triangle angle sum = 180, (each PROVED from above)
vertical angles equal, ...
rule of the game: any line may rest ONLY on the lines above it.Where one counterexample ends the argument
There is a sharp asymmetry between the two halves of a proof that beginners often miss. To establish that a general claim is true, no number of examples is ever enough — checking a thousand triangles does not prove every triangle obeys a rule. But to establish that a claim is false, a single counterexample is decisive and final. One triangle that breaks the pattern, and the claimed theorem is dead, no appeal. This is why mathematicians treat 'it works in every case I tried' as a hint, not a result.
A tiny worked example. Suppose someone claims 'every angle that looks like a right angle measures exactly 90 degrees.' To test it you don't gather agreeing pictures; you hunt for one that disagrees. Sketch an angle that looks square but is drawn at 88 degrees — done, the claim falls. Now flip it: to defend the real statement 'all right angles are equal,' you cannot sketch your way to certainty at all; you must invoke the postulate that says so. Examples can only ever destroy a claim, never build one.
- Name the four kinds: undefined term, definition, postulate, theorem.
- For any sentence, ask: may I use this freely, or must I prove it first?
- To prove a general claim TRUE, give a chain of reasoning — never just examples.
- To prove a claim FALSE, give exactly one counterexample.
Putting the pieces in motion
So the machine is now assembled: undefined terms give us pieces, definitions name useful arrangements, postulates and common notions supply the rules we agree to, and theorems are everything we can force out of those rules by honest reasoning. A proof is just a guided walk down the layers, where every step cites a sentence you are already entitled to. Nothing magical happens; the certainty comes entirely from being strict about which sentences you're allowed to lean on.
In the next guide we put real, measurable objects under this microscope — segments, rays, and angles you can actually assign a number to — and meet the addition postulates that let lengths and angle measures add up. After that, the two If-Then guides turn this whole apparatus loose and you'll write the two-column proof the rung is named for. You already have the hardest idea: in geometry, you don't believe a thing because it looks right. You believe it because the rules of the game force it.