MISSION · Geometry
Writing a Geometric Proof
The difference between "it looks true" and "it is true"
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A proof is a chain of statements, each one justified by something already known, ending at the thing you were asked to show.
Why measuring isn’t proving
Draw a triangle, measure its three angles, and they’ll total about \(180°\) — give or take your protractor. Draw a hundred more and you’ll have a hundred pieces of evidence and no proof.
A proof settles all triangles at once, including the ones nobody has drawn. That’s the trade: it costs more effort than measuring, and it buys certainty that no amount of measuring can.
It also means a single counterexample destroys a claim permanently. “All prime numbers are odd” dies at 2. Finding one counterexample is a complete disproof — which makes hunting for one the right first move when a claim smells wrong.
The shape of a proof
You are given some facts and asked to reach a conclusion. Every line in between is a statement plus the reason it holds, and the reasons come from a fixed stock: definitions, previously proved theorems, and the given facts.
A two-column layout makes the discipline visible:
| Statement | Reason |
|---|---|
| \(AB = CD\) | Given |
| \(\angle ABE = \angle CDE\) | Alternate angles, \(AB \parallel CD\) |
| \(\angle AEB = \angle CED\) | Vertically opposite angles |
| \(\triangle ABE \cong \triangle CDE\) | ASA |
| \(AE = CE\) | Corresponding parts of congruent triangles |
Nothing appears without a reason. “It looks equal” is not a reason, and neither is “from the diagram” — diagrams are sketches, not evidence.
The congruence tests
Most school proofs run through triangle congruence, so the four valid tests are the tools you’ll reach for:
- SSS — three pairs of sides equal.
- SAS — two sides and the angle between them.
- ASA — two angles and the side between them.
- RHS — right angle, hypotenuse, and one other side.
SSA is not a test. Two sides and a non-included angle can produce two different triangles — the same ambiguity that troubles the sine rule. And AAA only proves similarity, not congruence: same shape, any size.
Once two triangles are congruent, every remaining pair of parts is equal, which is usually the line that finishes the proof.
How to find the argument
Work from both ends. Write down what you’re given and what follows immediately. Then look at the conclusion and ask what would be enough to produce it. Usually the two searches meet in the middle, and the meeting point is a pair of congruent triangles.
Mark equal sides and angles on the diagram as you establish them. Half of proof-writing is bookkeeping, and a marked diagram does most of it for you.
Common mix-ups
- Assuming what the diagram appears to show. Only marked or given information counts.
- Using SSA, which isn’t a congruence test.
- Stating a conclusion with no reason. Every line needs one.
- Proving the wrong thing. Similar is not congruent.
Check yourself
Two triangles share a side, and you know two pairs of angles are equal. Which test applies, and what does the shared side give you?
The shared side is equal to itself — “common side” — so with two pairs of angles you have ASA (or AAS), and the triangles are congruent.
LEVEL 3 · REFLECT
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