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.

### Watch This

CK-12 Indirect Proof in Algebra and Geometry

James Sousa: Introduction to Indirect Proof

### Guidance

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
of the conclusion (second half) of the statement.*opposite* - Proceed as if this assumption is true to find the
*contradiction.* - Once there is a contradiction, the original statement is true.
Use variables so that the contradiction can be generalized.*DO NOT use specific examples.*

The easiest way to understand indirect proofs is by example.

#### Example A (Algebra Example)

If \begin{align*}x=2\end{align*}

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

**of "If \begin{align*}x=2\end{align*}**

*opposite*If \begin{align*}x=2\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*}

**and \begin{align*}3x - 5 \neq 10\end{align*}**

*assumption is incorrect***.**

*true*#### Example B (Geometry Example)

If \begin{align*}\triangle ABC\end{align*}

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*}** contradicts** the Triangle Sum Theorem that says the three angle measures of all triangles add up to \begin{align*}180^\circ\end{align*}

#### Example C (Geometry Example)

If \begin{align*}\angle {A} \end{align*}

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*}

CK-12 Indirect Proof in Algebra and Geometry

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### Guided Practice

1. If \begin{align*}n\end{align*}

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*}

**Answers:**

1. First, assume the ** opposite** of “\begin{align*}n\end{align*}

\begin{align*}n\end{align*}

Now, square \begin{align*}n\end{align*}

If \begin{align*}n\end{align*}

\begin{align*}n^2 = (2a)^2 = 4a^2\end{align*}

This means that \begin{align*}n^2\end{align*}** contradicts our assumption** that \begin{align*}n\end{align*}

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

**of "If \begin{align*}x=3\end{align*}**

*opposite*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

**and \begin{align*}4x+1 \neq 17\end{align*} is**

*assumption is incorrect***.**

*true*### Explore More

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 \begin{align*}\triangle ABC\end{align*} is equilateral, then the measure of the base angles cannot be \begin{align*}72^\circ\end{align*}.
- If \begin{align*}x=11\end{align*} then \begin{align*}2x-3\neq 21\end{align*}.
- If \begin{align*}\triangle ABC\end{align*} is a right triangle, then it cannot have side lengths 3, 4, and 6.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.8.