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Indirect Proof in Algebra and Geometry

Proof by contradiction.

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Indirect Proof in Algebra and Geometry

Indirect Proofs

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.

What if you wanted to prove a statement was true without a two-column proof? How might you go about doing so?

Examples

Example 1 (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 2 (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 3 (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*}.

Example 4

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.

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.

Example 5

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.”)

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.

Review

Prove the following statements true indirectly.

  1. 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.
  2. 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.
  3. If \begin{align*}x > 3\end{align*}, then \begin{align*}x^2 > 9\end{align*}.
  4. The base angles of an isosceles triangle are congruent.
  5. 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.
  6. 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.
  7. 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).
  8. If \begin{align*}\triangle ABC\end{align*} is equilateral, then the measure of the base angles cannot be \begin{align*}72^\circ\end{align*}.
  9. If \begin{align*}x=11\end{align*} then \begin{align*}2x-3\neq 21\end{align*}.
  10. If \begin{align*}\triangle ABC\end{align*} is a right triangle, then it cannot have side lengths 3, 4, and 6.

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.8. 

Resources

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Vocabulary

Indirect Proof or Proof by Contradiction

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.

indirect proof

An indirect proof takes the conclusion from a hypothesis and assumes it is false until a contradiction is reached, thus proving the original hypothesis is true.

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

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