What if you know something is true but cannot figure out how to prove it directly? After completing this Concept, you'll be able to indirectly prove a statement by way of contradiction.
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CK-12 Foundation: Chapter5IndirectProofA
James Sousa: Introduction to Indirect Proof
Guidance
Until now, we have proved theorems true by direct reasoning, where conclusions are drawn from a series of facts and previously proven theorems. However, we cannot always use direct reasoning to prove every theorem.
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 \begin{align*}x=2\end{align*}, then \begin{align*}3x - 5 \neq 10\end{align*}. 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 \begin{align*}x=2\end{align*}, then \begin{align*}3x - 5 \neq 10\end{align*}":
If \begin{align*}x=2\end{align*}, then \begin{align*}3x - 5 = 10\end{align*}.
Take this statement as true and solve for \begin{align*}x\end{align*}.
\begin{align*}3x - 5 &= 10\\ 3x &= 15\\ x &= 5\end{align*}
But \begin{align*}x = 5\end{align*} contradicts the given statement that \begin{align*}x = 2\end{align*}. Hence, our assumption is incorrect and \begin{align*}3x - 5 \neq 10\end{align*} is true.
Example B (Geometry Example)
If \begin{align*}\triangle ABC\end{align*} is isosceles, then the measure of the base angles cannot be \begin{align*}92^\circ\end{align*}. Prove this indirectly.
Remember, to start assume the opposite of the conclusion.
The measure of the base angles are \begin{align*}92^\circ\end{align*}.
If the base angles are \begin{align*}92^\circ\end{align*}, then they add up to \begin{align*}184^\circ\end{align*}. This contradicts the Triangle Sum Theorem that says the three angle measures of all triangles add up to \begin{align*}180^\circ\end{align*}. Therefore, the base angles cannot be \begin{align*}92^\circ\end{align*}.
Example C (Geometry Example)
If \begin{align*}\angle {A} \end{align*} and \begin{align*}\angle {B}\end{align*} are complementary then \begin{align*}\angle {A} \leq 90^\circ\end{align*}. Prove this by contradiction.
Assume the opposite of the conclusion.
\begin{align*}\angle {A} > 90^\circ\end{align*}.
Consider first that the measure of \begin{align*}\angle {B}\end{align*} cannot be negative. So if \begin{align*}\angle {A} > 90^\circ\end{align*} this contradicts the definition of complementary, which says that two angles are complementary if they add up to \begin{align*}90^\circ\end{align*}. Therefore, \begin{align*}\angle {A} \leq 90^\circ\end{align*}.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter5IndirectProofB
Vocabulary
An Indirect Proof or Proof by Contradiction is a method of proof where the conclusion from a hypothesis is assumed to be false (or opposite of what it states) and then a contradiction is reached from the given or deduced statements.
Guided Practice
1. If \begin{align*}n\end{align*} is an integer and \begin{align*}n^2\end{align*} is odd, then \begin{align*}n\end{align*} 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 \begin{align*}x=3\end{align*}, then \begin{align*}4x+1 \neq 17\end{align*}. Prove this statement is true by contradiction.
Answers:
1. First, assume the opposite of “\begin{align*}n\end{align*} is odd.”
\begin{align*}n\end{align*} is even.
Now, square \begin{align*}n\end{align*} and see what happens.
If \begin{align*}n\end{align*} is even, then \begin{align*}n = 2a\end{align*}, where \begin{align*}a\end{align*} is any integer.
\begin{align*}n^2 = (2a)^2 = 4a^2\end{align*}
This means that \begin{align*}n^2\end{align*} is a multiple of 4. No odd number can be divided evenly by an even number, so this contradicts our assumption that \begin{align*}n\end{align*} is even. Therefore, \begin{align*}n\end{align*} must be odd if \begin{align*}n^2\end{align*} 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 \begin{align*}x=3\end{align*}, then \begin{align*}4x+1 \neq 17\end{align*}":
If \begin{align*}x=3\end{align*}, then \begin{align*}4x+1=17\end{align*}
Take this statement as true and solve for \begin{align*}x\end{align*}.
\begin{align*}4x+1 &= 17\\ 4x &= 16\\ x &= 4\end{align*}
\begin{align*}x = 4\end{align*} contradicts the given statement that \begin{align*}x = 3\end{align*}. Hence, our assumption is incorrect and \begin{align*}4x+1 \neq 17\end{align*} is true.
Practice
Prove the following statements true indirectly.
- If \begin{align*}n\end{align*} is an integer and \begin{align*}n^2\end{align*} is even, then \begin{align*}n\end{align*} is even.
- If \begin{align*}m \angle A \neq m \angle B\end{align*} in \begin{align*}\triangle ABC\end{align*}, then \begin{align*}\triangle ABC\end{align*} is not equilateral.
- If \begin{align*}x > 3\end{align*}, then \begin{align*}x^2 > 9\end{align*}.
- The base angles of an isosceles triangle are congruent.
- If \begin{align*}x\end{align*} is even and \begin{align*}y\end{align*} is odd, then \begin{align*}x + y\end{align*} is odd.
- In \begin{align*}\triangle ABE\end{align*}, if \begin{align*}\angle A\end{align*} is a right angle, then \begin{align*}\angle B\end{align*} cannot be obtuse.
- If \begin{align*}A, \ B\end{align*}, and \begin{align*}C\end{align*} are collinear, then \begin{align*}AB + BC = AC\end{align*} (Segment Addition Postulate).
- If a collection of nickels and dimes is worth 85 cents, then there must be an odd number of nickels.
- Hugo is taking a true/false test in his Geometry class. There are five questions on the quiz. The teacher gives her students the following clues: The last answer on the quiz is not the same as the fourth answer. The third answer is true. If the fourth answer is true, then the one before it is false. Use an indirect proof to prove that the last answer on the quiz is true.
- On a test of 15 questions, Charlie claims that his friend Suzie must have gotten at least 10 questions right. Another friend, Larry, does not agree and suggests that Suzie could not have gotten that many correct. Rebecca claims that Suzie certainly got at least one question correct. If only one of these statements is true, how many questions did Suzie get right?
- If one angle in a triangle is obtuse, then each other angle is acute.
- If \begin{align*}3x+7\geq 13\end{align*}, then \begin{align*}x\geq 2\end{align*}.
- If segment AD is perpendicular to segment BC, then \begin{align*}\angle ABC\end{align*} is not a straight angle.
- If two alternate interior angles are not congruent, then the lines are not parallel.
- In an isosceles triangle, the median that connects the vertex angle to the midpoint of the base bisects the vertex angle.