What if you wanted to prove a statement was true without a two-column proof? How might you go about doing so? After completing this Concept, you'll be able to indirectly prove a statement by way of contradiction.
Most likely, the first type of formal proof you learned was a direct proof using direct reasoning. Most of the proofs done in geometry are done in the two-column format, which is a direct proof format. Another common type of reasoning is indirect reasoning, which you have likely done outside of math class. Below we will formally learn what an indirect proof is and see some examples in both algebra and geometry.
Indirect Proof or Proof by Contradiction: When the conclusion from a hypothesis is assumed false (or opposite of what it states) and then a contradiction is reached from the given or deduced statements.
In other words, if you are trying to show that something is true, show that if it was not true there would be a contradiction (something else would not make sense).
The steps to follow when proving indirectly are:
- Assume the opposite of the conclusion (second half) of the statement.
- Proceed as if this assumption is true to find the contradiction.
- Once there is a contradiction, the original statement is true.
- DO NOT use specific examples. Use variables so that the contradiction can be generalized.
The easiest way to understand indirect proofs is by example.
Example A (Algebra Example)
If , then . Prove this statement is true by contradiction.
Remember that in an indirect proof the first thing you do is assume the conclusion of the statement is false. In this case, we will assume the opposite of "If , then ":
If , then .
Take this statement as true and solve for .
But contradicts the given statement that . Hence, our assumption is incorrect and is true.
Example B (Geometry Example)
If is isosceles, then the measure of the base angles cannot be . Prove this indirectly.
Remember, to start assume the opposite of the conclusion.
The measure of the base angles are .
If the base angles are , then they add up to . This contradicts the Triangle Sum Theorem that says the three angle measures of all triangles add up to . Therefore, the base angles cannot be .
Example C (Geometry Example)
If and are complementary then . Prove this by contradiction.
Assume the opposite of the conclusion.
Consider first that the measure of cannot be negative. So if this contradicts the definition of complementary, which says that two angles are complementary if they add up to . Therefore, .
1. If is an integer and is odd, then is odd. Prove this is true indirectly.
2. Prove the SSS Inequality Theorem is true by contradiction. (The SSS Inequality Theorem says: “If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle's two congruent sides is greater in measure than the included angle of the second triangle's two congruent sides.”)
3. If , then . Prove this statement is true by contradiction.
1. First, assume the opposite of “ is odd.”
Now, square and see what happens.
If is even, then , where is any integer.
This means that is a multiple of 4. No odd number can be divided evenly by an even number, so this contradicts our assumption that is even. Therefore, must be odd if is odd.
2. First, assume the opposite of the conclusion.
The included angle of the first triangle is less than or equal to the included angle of the second triangle.
If the included angles are equal then the two triangles would be congruent by SAS and the third sides would be congruent by CPCTC. This contradicts the hypothesis of the original statement “the third side of the first triangle is longer than the third side of the second.” Therefore, the included angle of the first triangle must be larger than the included angle of the second.
3. In an indirect proof the first thing you do is assume the conclusion of the statement is false. In this case, we will assume the opposite of "If , then ":
If , then
Take this statement as true and solve for .
contradicts the given statement that . Hence, our assumption is incorrect and is true.
Prove the following statements true indirectly.
- If is an integer and is even, then is even.
- If in , then is not equilateral.
- If , then .
- The base angles of an isosceles triangle are congruent.
- If is even and is odd, then is odd.
- In , if is a right angle, then cannot be obtuse.
- If , and are collinear, then (Segment Addition Postulate).
- If is equilateral, then the measure of the base angles cannot be .
- If then .
- If is a right triangle, then it cannot have side lengths 3, 4, and 6.