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USA - Geometry Proof and reasoning

Proofs by induction and contradiction

20 practice questions 2 video lessons Theory + worked examples

Proof by Induction and Contradiction

California Geometry • G-CO • Proof & Reasoning

Proof by Induction and Contradiction is a topic in Proof & Reasoning in the California Common Core State Standards. It is aligned to the G-CO strand of the framework, which requires students to construct logical arguments, including indirect proofs.

Proof by contradiction assumes the negation of a claim and derives an absurdity, while induction proves a statement across an ordered chain of cases.

California Geometry › Proof & Reasoning › Proof by Induction and Contradiction  —  G-CO

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  • Proof by Contradiction | ExamSolutions Watch
  • Proof by induction | Sequences, series and induction | Precalculus | Khan Academy Watch
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Theory

Beyond direct proof, two powerful techniques appear in geometry:

  • Proof by contradiction: assume the statement is false, reason logically until you reach an impossibility, and conclude the assumption — and so the statement — must be the opposite.
  • Proof by induction: prove a base case (usually \(n=1\)), then the inductive step (if true for \(k\), it is true for \(k+1\)). Together these prove it for all \(n\).
Contradiction proves one statement; induction proves a whole family indexed by \(n\).
Proof by contradiction Proof by contradiction Proof by contradiction 1. assume the opposite is true 2. reason until something impossible 3. the assumption must be false
Proof by contradiction: assume the opposite, reach an impossibility.
Proof by induction Proof by induction Proof by induction base case: show it holds for n = 1 inductive step: if true for k, true for k+1 therefore true for all n
Proof by induction: base case plus inductive step.

The two structures:

\[\text{contradiction: assume } \lnot P\Rightarrow \text{impossible}\Rightarrow P\]
\[\text{induction: } P(1)\ \text{and}\ \big(P(k)\Rightarrow P(k+1)\big)\Rightarrow P(n)\ \forall n\]
contradiction assumes not P and reaches an impossibility; induction proves a base case and an inductive step
Induction needs both parts: the base case and the step. Either alone proves nothing.

How to use each technique

  1. Contradiction: assume the negation, derive an impossibility, conclude the statement.
  2. Induction — base case: verify \(P(1)\).
  3. Induction — step: assume \(P(k)\), prove \(P(k+1)\).
  4. Conclude the statement holds for every \(n\).
Example 1 — Set up a contradiction
To prove “a triangle has at most one right angle” by contradiction, what do you assume?
Solution

Assume the opposite: the triangle has two right angles. Then those two angles alone total \(180^\circ\), leaving nothing for the third — impossible.

\(90^\circ+90^\circ\)\(=\)\(180^\circ\ (\text{no third angle})\)
assume two right angles, which already sum to 180 degrees
Example 2 — The contradiction
After reaching an impossible result in a proof by contradiction, what do you conclude?
Solution

The assumption was false, so the original statement is true.

the assumption is false, so the statement is true
Example 3 — Base case
In a proof by induction, what is the base case?
Solution

Showing the statement is true for the first value, usually \(n=1\).

the base case shows it is true for n equals 1
Example 4 — Inductive step
What does the inductive step assume and show?
Solution

It assumes the statement holds for \(n=k\) and shows it then holds for \(n=k+1\).

\(\text{true for } k\)\(\Rightarrow\)\(\text{true for } k+1\)
assume true for k and show true for k plus 1

Common pitfalls

In contradiction, assume the exact opposite, then look for the impossibility.
Induction needs the base case AND the step. Skipping the base case leaves the chain unanchored.
The inductive step assumes \(P(k)\) — that assumption is allowed; it is not circular.

Frequently asked questions

How does proof by contradiction work?

Assume the statement is false, reason until you reach something impossible, and conclude the statement must be true.

What are the two parts of a proof by induction?

The base case (true for the first value) and the inductive step (if true for \(k\), then true for \(k+1\)).

What is the base case?

The starting value, usually \(n=1\), where you show the statement holds.

Why do you assume the opposite in a contradiction proof?

Because deriving an impossibility from that assumption shows the assumption is false, which proves the original statement.

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