A youtube viewer of mine requested a video on **how to write an indirect proof**. Before making the video, I thought it would be good to write a blog post about this topic before I make my video, because an indirect proof is best written as a paragraph proof. There are three steps to writing an indirect proof.

- Assume that the conclusion is false, by negating the prove statement.
- Establish that the assumption in step #1 leads to a contradiction of some fact i.e. definition, postulate, corollary or theorem.
- State the assumption must be false, thus, the conclusion or prove statement is true.

Steps 1 and 2 involve all of the thought and memory skills and can be discussed separately. Do not make this harder than it is. Step 1 involves writing the negation of a statement. Step 2 requires you to pull on your knowledge of geometric definitions, postulates, theorems and corollaries to recognize the contradiction between a known geometric fact and the assumption in step 1. Step 3 involves stating the obvious: Since the assumption is false, the prove statement must be true. If you are confused, check out the examples.

**Example 1 – Prove the Exterior Angle Inequality Theorem with Indirect Proof**

Given: is an exterior angle of

Prove:

Figure 1 - Indirect Proof Diagram

Step 1 – Assume that , this means that .

Step 2 – We need to establish that contradicts a mathematical fact.

gives two different situations that need to be tested:

or .

By the Exterior Angle Theorem, and using substitution, . Subtracting from both sides gives . This contradicts the fact an angle must have a measure greater than 0.

By the Exterior Angle Theorem, Angles must have a positive measure, the definition means and .

Step 3 – In each instance, the assumption from step 1 is contradicted of a know mathematical fact. Thus, the assumption that is false. So, the original prove statement, , must be true.

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Filed under: Exterior Angle Inequality Theorem, Geometry, Indirect Proofs, Proofs Tagged: | Geometry, How to Write an Indirect Proof, Indirect Proof, Writing an Indirect Proof

Guillermo Bautista, on January 10, 2010 at 10:20 pm said:Mr. Pi,

I also have a draft of a blog post on Proof by Contradiction. Maybe I’ll link this one.

Best,

Guillermo

Mr. Pi, on January 12, 2010 at 9:52 pm said:Guillermo,

I look forward to reading reading your post on indirect proof. I like your use of Proof by Contradiction. This name gives a more accurate description as what needs to be done.

Regards,

Mr. Pi