How to Write a Proof — Print
A proof is not a calculation. It is a chain of statements where every link carries a reason.
What the practice questions cover
These questions use 17 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Base — The number being multiplied. In 53 the base is 5.
- Scale — What one grid step is worth — and where the axis starts.
- Converse — If a2 + b2 = c2, then the triangle must be right-angled.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Transversal — A line that crosses two or more other lines.
- Corresponding angles — Same position at each crossing. Equal when the lines are parallel.
- Alternate interior angles — Between the parallels, on opposite sides of the transversal. Equal.
- Midpoint — The point halfway between two others: average the x values and average the y values.
- Parallel — Same slope, different intercept.
- Segment — Two endpoints and everything between them. Written AB.
- 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.
- Counterexample — A single case that makes a claim false. One is enough.
- CPCTC — Corresponding Parts of Congruent Triangles are Congruent — used after proving congruence.
- Reflexive property — Anything is congruent to itself. A shared side is congruent to itself.
- Rhombus — A parallelogram with four congruent sides.
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?
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.
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.
- M is the midpoint of AB — Given Always open with what you were handed.
- AM = MB — definition of midpoint The definition is doing the work.
- AM + MB = AB — segment addition postulate M lies between A and B.
- AM + AM = AB — substitution Replace MB using line 2.
- 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."
- One counterexample is enough. Disproving never needs a general argument.
- Take a kite with sides 5, 5, 8, 8. Two pairs of equal sides ✓
- But in a kite the equal sides are adjacent, not opposite. So no pair of sides is parallel.
- 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?
- The converse swaps the parts: "if two angles are congruent, they are vertical." Write it out before judging it.
- Look for a counterexample. Two angles can be congruent for many reasons.
- The two base angles of an isosceles triangle are congruent — and nowhere near vertical. They do not even share a vertex.
- 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.
Problem 1
What is always the first line of a two-column proof?
- The given
- What you are proving
- A definition
- A theorem
Worked answer
The given
The given is your starting material.
The prove statement is the last line.
Problem 2
A statement in a proof with no reason beside it is
- not acceptable, even if true
- fine if obvious
- acceptable once per proof
- a definition
Worked answer
not acceptable, even if true
Every statement needs a justification.
A true but unjustified line is still a gap.
Problem 3
Write the converse of "if it rains, the ground is wet".
Worked answer
if the ground is wet, it rains
Original: if P then Q
Converse: if Q then P
Problem 4
How many counterexamples are needed to disprove a claim?
Worked answer
1
One case that fails makes the claim false.
Proving needs every case; disproving needs one.
Problem 5
You know M is the midpoint of AB. What reason lets you write AM = MB?
Worked answer
definition of midpoint
A midpoint cuts a segment into two congruent parts.
That is the definition.
Problem 6
Is the converse of "if two angles are vertical, they are congruent" true? Answer yes or no.
Worked answer
no
Converse: if two angles are congruent, they are vertical.
The base angles of an isosceles triangle are congruent but not vertical.
So the converse is false.
Problem 7
In a proof, two triangles share a side. What reason lets you say that side is congruent to itself?
Worked answer
reflexive property
BD ≅ BD
That is the reflexive property.
Problem 8
CPCTC may be used
- only after proving two triangles congruent
- at any point in a proof
- to prove triangles congruent
- only in coordinate proofs
Worked answer
only after proving two triangles congruent
It is what congruence gives you.
Using it to prove congruence would be circular.
Problem 9
Give a counterexample to "all quadrilaterals with four equal sides are squares".
Worked answer
a rhombus
A rhombus has four equal sides.
Its angles need not be right angles.
Problem 10
You are given that AB ∥ CD. What angle relationship does a transversal give you?
Worked answer
congruent alternate interior angles
Alternate interior angles are congruent.
Corresponding angles are congruent too.
Problem 11
If "if P then Q" is true, which of these is also always true?
- If not Q then not P
- If Q then P
- If not P then not Q
- None of these
Worked answer
If not Q then not P
The contrapositive is always true with the original.
The converse and inverse are not.
Problem 12
A proof says "∠ A is a right angle" because the diagram looks square. Is that valid? Answer yes or no.
Worked answer
no
Diagrams are not drawn to scale.
A right angle must be given or marked.
Problem 13
The converse of the Pythagorean theorem is true. What does it let you conclude?
Worked answer
the triangle is right-angled
If a2 + b2 = c2 with c longest, then the triangle has a right angle.
It is a genuine theorem, proved separately.
Problem 14
Write the contrapositive of "if a shape is a square, then it has four right angles".
Worked answer
if it does not have four right angles, it is not a square
Original: if P then Q
Contrapositive: if not Q then not P
Problem 15
In a proof you want to show AD ≅ CD. What is usually the step before?
Worked answer
prove the triangles containing them are congruent
Prove △ ABD ≅ △ CBD.
Then CPCTC gives the segments.
Problem 16
Disprove: "if two triangles have the same perimeter, they are congruent."
Worked answer
a 3-4-5 and a 4-4-4 triangle both have perimeter 12
3-4-5 has perimeter 12.
So does 4-4-4.
Different shapes, same perimeter.