Prove something is true by assuming it's false — and showing that assumption breaks the rules.
The Mnemonic
Proving Something True By Proving It Can't Be False
Most proofs start from what's given and build directly toward the conclusion. Proof by contradiction — also called indirect proof — does the opposite: it starts by assuming the conclusion is false, then shows that assumption leads somewhere impossible. If assuming "false" breaks the rules, the only option left is that the original statement must actually be true.
This isn't a trick specific to geometry — it's a general logic technique borrowed into geometric proofs, used specifically when a direct proof is awkward or when you're trying to prove something can't happen (uniqueness statements, "no two lines...", "there is only one...").
The four-step shape of every indirect proof — the same pattern no matter what you're proving
💡 Memory Trick
"Indirect proof (proof by contradiction): assume opposite is true → reach a contradiction." Four moves, always in this order: assume the opposite, reason forward with valid logic, land on something false or impossible, then conclude the original statement was true all along.
Why It Works
The Logic That Makes This Valid
This method leans on a basic rule of logic: any statement is either true or false — there's no third option. If you can show that "false" leads to an impossible result, "false" must be wrong, which leaves "true" as the only remaining possibility.
It feels indirect because you never touch the actual statement you're proving until the very last line — you spend almost the whole proof reasoning about its opposite instead. That's exactly why it's called an indirect (or contrapositive-style) proof, as opposed to a direct proof that reasons straight from given to conclusion.
Using It In A Proof
When to Reach for This Method
Proof by contradiction isn't the default — most proofs should still be attempted directly first. But certain phrasings are strong signals that indirect proof is the faster path.
1
Watch for "no" and "only one" statements
Claims like "there is no triangle with two right angles" or "a line has only one perpendicular through a given point" are hard to prove directly, but very natural to attack by assuming the opposite.
"No triangle has two right angles" becomes: assume a triangle DOES have two right angles, then show the three angles would sum to more than 180° — impossible.
2
State the assumption clearly
Write the negation of the conclusion as your starting assumption — this becomes the very first line of the proof, explicitly labeled as an assumption, not a given fact.
3
Reason forward until something breaks
Use definitions, postulates, and theorems you already know to be true, and follow the assumption's logical consequences until you contradict a known fact, a given piece of information, or the assumption itself.
Full Worked Example
Proving a Triangle Can't Have Two Right Angles
Prove: No triangle can have two right angles.
1
Assume the opposite
Assume, for the sake of contradiction, that a triangle DOES exist with two right angles — call them ∠A = 90° and ∠B = 90°.
2
Reason forward using a known theorem
By the Triangle Angle Sum Theorem, ∠A + ∠B + ∠C = 180°. Substituting the assumed values: 90° + 90° + ∠C = 180°.
3
Reach the contradiction
Solving gives ∠C = 0°. But an angle of 0° means there's no actual triangle — two of the sides would have to lie exactly on top of each other. This directly contradicts the definition of a triangle as a closed figure with three distinct sides and angles.
4
Conclude the original statement is true
Since assuming "a triangle has two right angles" leads to an impossible triangle, the assumption must be false. Therefore, no triangle can have two right angles.
Notice the proof never directly manipulated "no triangle has two right angles" — it only worked with its opposite until that opposite broke down.
🎯 Quick Worked Example
Prove: A triangle cannot have two obtuse angles (angles greater than 90°).
1
Assume the opposite. Suppose a triangle DOES have two obtuse angles, both greater than 90° — say ∠A > 90° and ∠B > 90°.
2
Reason forward. Then ∠A + ∠B is already greater than 180°, before even adding the third angle ∠C.
3
Reach the contradiction. But the Triangle Angle Sum Theorem requires ∠A + ∠B + ∠C = 180° exactly — with ∠A + ∠B already over 180°, ∠C would have to be negative, which is impossible. So the assumption is false, and a triangle cannot have two obtuse angles.
📌 Exam Application
When an exam asks you to prove a statement containing the words "no," "never," "cannot," "only one," or "unique," try proof by contradiction before spending time hunting for a direct approach — these phrasings are the clearest signal this method applies, and forcing a direct proof on them is often far harder than it needs to be.
⚠️ Most Common Proof by Contradiction Mistakes
Trap 1 — Forgetting to explicitly state the assumption: The proof must clearly label its very first line as an assumption ("Assume, for the sake of contradiction, that...") — skipping this makes the logic of the whole proof unclear to a grader, even if your math is correct.
Trap 2 — Stopping before an actual contradiction: Reaching an answer that just seems "weird" or "unlikely" isn't enough — you need a genuine logical impossibility (like a negative angle, an angle of 0° in a triangle, or a direct contradiction of a given fact). A merely surprising result is not a contradiction.
✓ Quick Self-Test
1) What is the first step of a proof by contradiction? 2) Why does reaching an impossible result prove the original statement true? 3) What phrasing in a problem statement suggests proof by contradiction might be the right approach? 4) In the two-right-angles example, what specific impossible result was reached? 5) Is proof by contradiction typically the first method you should try, or a fallback?