How to Write a Proof

A proof is not a calculation. It is a chain of statements where every link carries a reason.

Claim: the three angles of any triangle add to 180°. How would you convince someone?

You could draw three triangles and measure. All three come to 180°. Case closed?

why should anyone believe you? measuring I measured three triangles and each came to 180° not a proof proving given three angles because parallel lines so they make a line therefore 180° always covers every triangle three examples cannot rule out a fourth; a reason can

No — because there are infinitely many triangles, and you checked three. A fourth could still misbehave.

Now try the other column. Draw a line through the top vertex parallel to the base.

Two of the angles reappear at the top as alternate angles, sitting alongside the third. Together they make a straight line.

That argument never mentions a particular triangle, so it covers every triangle at once. That is what a proof buys you, and why measuring cannot buy it.

every two-column proof has the same shape STATEMENT REASON what you were told Given a step that follows a definition another step a theorem what you wanted follows from above the left column may never get ahead of the right one a statement with no reason beside it is a guess, however true it is

What a proof has to do

A proof starts from things already agreed — the given, definitions, postulates and earlier theorems — and reaches the required statement by steps nobody can object to.

Every statement needs a reason beside it. A statement with no reason is a guess, even when it happens to be true.

And the reason must already be available: you cannot use a theorem you have not proved yet.

The test

Hand your proof to someone who disagrees with the conclusion. If they cannot point at a single line and say "why?", you have proved it.

The two-column format

Statements go on the left, reasons on the right, one pair per line. The first line is always the given; the last is always what you were asked to prove.

The format is not the point — it is scaffolding that stops you skipping steps. A paragraph proof saying the same things is just as valid.

But while you are learning, the columns are worth keeping, because they make a missing reason impossible to hide.

Working backwards from the goal

Do not start at the top and hope. Start at the bottom and ask: what would be enough to give me this?

To prove two segments congruent, it would be enough to have two congruent triangles containing them. To get congruent triangles, I need three matching parts.

Keep asking until you hit something you were given. Then write the whole chain out forwards.

Mark the diagram first

Tick every given onto the picture before writing anything — equal sides, equal angles, right angles. The proof is usually visible once the diagram is marked.

Converse, inverse, contrapositive

If a statement says "if P then Q", its converse says "if Q then P" — and the converse can easily be false.

"If it is a square, it has four right angles" is true. Its converse, "if it has four right angles it is a square", is false — a rectangle is a counterexample.

The contrapositive, "if not Q then not P", is always true whenever the original is. That makes it a legitimate move in a proof; the converse is not.

The most common logical slip

Using a converse as though it followed automatically. It does not. Some converses are true and separately proved — the converse of Pythagoras, for instance — but each one needs earning.

Disproving something

To show a claim is false you do not need an argument. You need one counterexample.

"All quadrilaterals with four equal sides are squares" dies the moment you draw a rhombus that is not a square.

This asymmetry is worth noticing: proving needs every case, disproving needs one.

Worked examples

A short two-column proof

Given: M is the midpoint of AB. Prove: AM = 12 AB.

  1. M is the midpoint of AB — Given Always open with what you were handed.
  2. AM = MB — definition of midpoint The definition is doing the work.
  3. AM + MB = AB — segment addition postulate M lies between A and B.
  4. AM + AM = AB — substitution Replace MB using line 2.
  5. 2AM = AB, so AM = 12 AB — algebra The required statement, with every line justified.

Proved: AM = 12 AB

Finding a counterexample

Disprove: "if a quadrilateral has two pairs of equal sides, it is a parallelogram."

  1. One counterexample is enough. Disproving never needs a general argument.
  2. Take a kite with sides 5, 5, 8, 8. Two pairs of equal sides ✓
  3. But in a kite the equal sides are adjacent, not opposite. So no pair of sides is parallel.
  4. A kite is therefore a counterexample, and the claim is false. The true version requires the equal sides to be opposite.

A kite disproves it

Spotting a bad converse

"If two angles are vertical, they are congruent." Is the converse true?

  1. The converse swaps the parts: "if two angles are congruent, they are vertical." Write it out before judging it.
  2. Look for a counterexample. Two angles can be congruent for many reasons.
  3. The two base angles of an isosceles triangle are congruent — and nowhere near vertical. They do not even share a vertex.
  4. So the converse is false. The original statement is still perfectly true.

No — the converse is false

Words to know

Given
What the problem hands you. Always the first line of a proof.
Prove
The statement you must arrive at. Always the last line.
Two-column proof
Statements on the left, the reason for each on the right.
Converse
A statement with the if and the then swapped. It may or may not be true.
Counterexample
A single case that makes a claim false. One is enough.

Mistakes to watch for

Writing a statement with no reason beside it. That is the one thing a proof may never contain, because an unjustified line is exactly what a sceptical reader will stop at — and a true statement is not the same as a proved one.

Using the converse of a theorem as if it followed for free. "If P then Q" does not give you "if Q then P": four right angles do not make a square. Some converses are true, but each one has to be proved separately.

Assuming from the picture. A diagram may look like it has a right angle or a midpoint, but unless it is marked or given, you may not use it. Diagrams are drawn to help, not to scale.

Proving by example. Three triangles that each total 180° prove nothing about the fourth. Disproving takes one case; proving takes an argument that covers them all.

The Greeks were the first to insist on proof. Babylonian and Egyptian mathematics was a collection of recipes that worked; nobody asked why.

Thales of Miletus, around 600 BC, is credited with the first recorded proof — that a circle is bisected by its diameter.

It looks almost too obvious to bother with. That is the point: he was demonstrating what it meant to demonstrate anything.

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