What a Proof Is, and What a Definition Is
A geometric proof is a chain of claims in which every link carries a reason. Learning the form is most of the work, because once the form is automatic the geometry has room to happen.
Write down what you think the definition of a rectangle is. Now count how many separate facts you used. Could you remove one of them and still be describing only rectangles? Try it. If removing a fact lets in some other shape, that fact was necessary; if it does not, your definition had more in it than it needed.
A formal proof is a sequence of statements in which every statement carries a reason, and the reasons are only definitions, given information, or theorems already proved. What the diagram looks like is never a reason. A proof is judged on its chain, not on its conclusion.
$$\text{Given} \;\longrightarrow\; \text{statement + reason} \;\longrightarrow\; \cdots \;\longrightarrow\; \text{Conclusion}$$
Marks on a diagram ARE given information and may be used. Appearances are not: two sides that look equal, an angle that looks like a right angle, three points that look collinear. The discipline that separates a proof from a description is asking, of every line you write, "which definition, given fact or theorem allows this?"
Know
- That every statement in a formal proof requires a reason, and what counts as one
- That a definition is the minimum information needed to identify a figure
- The standard layout: Given, To prove, Proof, Conclusion
Understand
- Why the appearance of a diagram can never justify a step
- Why a definition with redundant conditions is still correct but less useful
Can Do
- Set out a short proof in the standard form with reasons
- Distinguish given information from assumed information in a diagram
- Test whether a proposed definition is minimal, by removing a condition and looking for a counterexample
Every formal proof has the same four parts, and writing them down in order does much of the thinking for you.
Given. List exactly what you are entitled to use, including everything marked on the diagram.
To prove. State the target. Writing it down stops you proving something adjacent by accident.
Proof. A numbered or line-by-line chain. Each line is a statement, and beside it, in brackets, the reason.
Conclusion. State the result in the words the question used.
A worked example of the form, on the simplest possible content:
| Statement | Reason |
|---|---|
| $\angle ABD = \angle DBC$ | given (marked on the diagram) |
| $\angle ABD + \angle DBC = \angle ABC$ | adjacent angles at $B$ |
| $\angle ABC = 2 \times \angle ABD$ | substituting the first line into the second |
Only three things may justify a line.
Given information. Anything stated in the question, and anything marked on the diagram: equal ticks, equal angle arcs, right-angle squares, parallel arrows.
A definition. "Opposite sides are parallel, by the definition of a parallelogram."
A theorem already proved. "Angle sum of a triangle", "vertically opposite angles are equal", "base angles of an isosceles triangle are equal".
And what does not count, however obvious it looks:
When two figures correspond, the ORDER of the letters carries the correspondence. Writing $\triangle ABC \equiv \triangle PQR$ says four things at once: $A$ matches $P$, $B$ matches $Q$, $C$ matches $R$, and therefore every matching side and angle is equal.
So from $\triangle ABC \equiv \triangle PQR$ you may immediately write $AB = PQ$, $\angle B = \angle Q$, $CA = RP$, and so on, each with the reason matching sides of congruent triangles.
Write the letters in the wrong order and every conclusion you draw is wrong, even though the triangles really are congruent. The statement $\triangle ABC \equiv \triangle QPR$ claims $A$ matches $Q$, which may simply be false.
A definition should contain exactly enough information to identify the figure: remove any part and something else gets in.
Take the rhombus. "A quadrilateral with four equal sides" is a definition: nothing but a rhombus has four equal sides. Remove the word "four" and a kite gets in.
Now consider "a quadrilateral with four equal sides and opposite sides parallel". That is true of every rhombus, but the second condition is redundant, because four equal sides already forces the parallels. It is a correct description and a poor definition.
Why the difference matters: every condition in a definition is something you must check when testing whether a figure qualifies, and something you may use once it does. A minimal definition gives the shortest test and the clearest proofs.
| Figure | Minimal definition | Remove a condition and... |
|---|---|---|
| Isosceles triangle | a triangle with two equal sides | any triangle qualifies |
| Rectangle | a parallelogram with one right angle | any parallelogram qualifies |
| Square | a rectangle with two adjacent sides equal | any rectangle qualifies |
Watch Me Solve It · 3 examples
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1List the given information$AB = AC \quad \text{(given)}$$\angle BAD = \angle CAD \quad \text{(given: } AD \text{ bisects } \angle BAC)$Everything you are entitled to use, and nothing else.
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2Find the shared element$AD = AD \quad \text{(common side)}$A side shared by both triangles is equal to itself; this is nearly always the third fact.
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3Name the congruence in matching order$\triangle ABD \equiv \triangle ACD \quad \text{(SAS)}$Two sides and the INCLUDED angle. $A$ matches $A$, $B$ matches $C$, $D$ matches $D$.
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4Draw the conclusion$\angle ADB = \angle ADC \quad \text{(matching angles of congruent triangles)}$The matching order established in the previous line is what makes this legitimate.
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1Separate the two claims$PQ \parallel SR \quad \text{(marked, so given)}$$PQ = SR \quad \text{(NOT marked)}$Only one of the two has a source.
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2Ask what the unmarked claim rests on$\text{appearance only}$A diagram is a sketch. Unmarked equality is not data.
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3Test whether it matters$\text{one pair of parallel sides alone gives a TRAPEZIUM}$A trapezium satisfies everything actually given, so the conclusion does not follow.
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4State the fault precisely$\text{the second premise is assumed, not given}$The reasoning from the two premises is fine; one premise is unavailable.
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1Check that it is correct$\text{every square satisfies it; nothing else does}$Correctness first: a definition that lets other shapes in is worse than a redundant one.
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2Remove one condition and look for an intruder$\text{four equal sides only} \Rightarrow \text{rhombus}$$\text{four right angles only} \Rightarrow \text{rectangle}$Both removals let something in, so neither condition can simply be deleted.
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3Weaken rather than delete$\text{a rectangle with two adjacent sides equal}$One right angle plus the parallelogram property forces the other three, and one pair of adjacent equal sides forces all four.
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4Conclude$\text{correct but not minimal}$Four right angles is three more than needed once the figure is known to be a parallelogram.
Brain Trainer · 4 problems
Four quick problems. Work each one, then reveal the answer.
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1 Is "the two sides look the same length" a valid reason in a proof?
A diagram is a sketch and is not drawn to scale.No -
2 Is "$AD = AD$, common side" a valid step?
A shared side is equal to itself, and this is a standard reason.Yes -
3 From $\triangle ABC \equiv \triangle PQR$, which side equals $BC$?
$B$ matches $Q$ and $C$ matches $R$.$QR$ -
4 Remove "two equal sides" from the definition of an isosceles triangle. What gets in?
Nothing is left but "a triangle".Every triangle
Multiple Choice · 5 questions
In a formal geometric proof, every statement must be accompanied by:
Which of these may be used as given information?
If $\triangle ABC \equiv \triangle PQR$, then $\angle C$ equals:
A definition is described as MINIMAL when:
A student proves a triangle is isosceles, and one of their reasons is "because the triangle is isosceles". This argument is:
Short Answer · 3 questions
(a) List the given information, including anything that follows from the word "midpoint".
(b) Prove that $\triangle PQM \equiv \triangle PRM$, giving a reason for every line.
(c) State one further conclusion that follows, with its reason.
"$AB = AC$ (given), $\angle B = \angle C$ (base angles of an isosceles triangle), $\angle A = 60°$ (it looks like an equilateral triangle), therefore $\triangle ABC$ is equilateral."
(a) Which lines are legitimate?
(b) Which is not, and why?
(c) What would need to be added to the given information for the conclusion to follow?
(b) A student defines a rectangle as "a quadrilateral with four right angles and opposite sides equal". Show this is correct but not minimal, and give a minimal version.
(c) Explain why a minimal definition is more useful in proofs than a full list of properties.
(b) A quadrilateral has one pair of opposite sides equal AND that same pair parallel. Must it be a parallelogram?
(c) A quadrilateral has one pair of opposite sides equal and the OTHER pair parallel. Must it be a parallelogram? Draw or describe a figure that settles the question.
Every line
A statement and a reason
Reasons
Given, definition, or proved theorem
Never
How the diagram looks
Definition
The minimum that identifies the figure
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