Indirect proof, also called proof by contradiction, is a powerful method used when a direct proof is difficult to write. In geometry, it helps prove statements about angles, triangles, parallel lines, and congruence by showing that the opposite claim cannot be true. The method is based on the law of excluded middle: a statement is either true or false.
If assuming the statement is false leads to an impossible result, then the statement must be true.
To write an indirect proof, begin by clearly stating what you want to prove, then assume its negation. Use accepted definitions, theorems, algebra, and given information to follow the consequences of that assumption. When the reasoning produces a contradiction, such as two angles adding to both 180° and 200°, the assumption must be rejected.
The conclusion is that the original statement is true.
Key Facts
- Indirect proof structure: assume not P, derive a contradiction, conclude P is true.
- Negation changes the statement to its logical opposite, such as x = 5 becoming x ≠ 5.
- Triangle angle sum theorem: A + B + C = 180°.
- If a contradiction follows from an assumption, the assumption is false.
- Common contradiction form: a value must equal two different numbers, such as x = 40 and x = 50.
- In geometry, contradictions often involve impossible angle sums, violated congruence conditions, or conflicting parallel line relationships.
Vocabulary
- Indirect proof
- A proof method that shows a statement is true by proving that its opposite leads to a contradiction.
- Contradiction
- A result that conflicts with a known fact, given statement, definition, or earlier proven theorem.
- Negation
- The logical opposite of a statement, such as changing greater than to less than or equal to.
- Assumption
- A statement temporarily accepted as true in order to explore its logical consequences.
- Theorem
- A mathematical statement that has already been proven and can be used to justify steps in a proof.
Common Mistakes to Avoid
- Assuming what you are trying to prove, instead of assuming its opposite. This is wrong because indirect proof must begin with the negation of the desired conclusion.
- Writing an unclear negation, such as changing x > 3 to x < 3 instead of x ≤ 3. This is wrong because the negation must include every case where the original statement is false.
- Stopping after finding an unusual result rather than a true contradiction. A contradiction must directly conflict with a definition, theorem, given fact, or previous conclusion.
- Forgetting the final conclusion after the contradiction is reached. The proof is incomplete unless you state that the original statement must be true.
Practice Questions
- 1 In triangle ABC, m∠A = 80° and m∠B = 65°. Use an indirect proof to show that m∠C cannot be 40°.
- 2 Two lines l and m are cut by a transversal. A pair of corresponding angles measure 3x + 10 and 5x - 30 degrees. If l is parallel to m, use contradiction to show x cannot equal 25.
- 3 Explain why an indirect proof is useful for proving that a triangle cannot have two obtuse angles.