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.
Understanding Geometry: Indirect Proof
The most important skill in an indirect proof is writing the correct assumption. The opposite of a statement is not always made by simply adding the word not. For example, the opposite of a claim that two lines are parallel is that they are not parallel.
The opposite of a claim that an angle is acute is that its measure is greater than or equal to ninety degrees. An acute angle cannot be exactly ninety degrees, so that boundary matters.
When a statement says every triangle has a property, its opposite says that at least one triangle fails to have that property. Careful negation prevents a proof from starting on the wrong path.
A contradiction must come from facts that are already secure. These can include givens, definitions, postulates, previously proved theorems, or results obtained through valid algebra. Suppose a proof needs to show that a triangle has at most one right angle.
Assume it has two right angles. Those two angles already total one hundred eighty degrees. The third interior angle would need to measure zero degrees because a triangle totals one hundred eighty degrees.
A triangle cannot have an interior angle of zero degrees. That impossible result does not mean that geometry is broken. It shows that the temporary assumption created the problem.
Indirect reasoning is useful when a diagram hides the most direct route. A diagram may suggest that two segments do not intersect, but drawings are not proof. Instead, assume that they do intersect at a point.
The intersection can create smaller triangles, angle pairs, or distance relationships. Known facts may then force one point to lie in two different places or force two distinct lines to share more than one point. In coordinate geometry, an assumption may lead to a slope being both a certain number and a different number.
In constructions, it may force a point to be both inside and outside a circle. These are precise conflicts, not merely results that seem unlikely from the picture.
Students should keep the assumed claim visible while writing each step. This makes it easier to check whether every deduction truly depends on the assumption or simply repeats a fact that was known before. State the exact contradiction near the end, such as an angle measure being both less than ninety degrees and at least ninety degrees.
Do not write only that this is impossible. Name the theorem or definition that makes it impossible.
Then reject only the assumption that caused the conflict. This method builds strong habits for later algebra, where it can prove that a number is irrational, establish inequalities, or show that an equation has no solution under certain conditions.
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.