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## How to Write Indirect Proofs – Exterior Angle Inequality Theorem

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.

1. Assume that the conclusion is false, by negating the prove statement.
2. Establish that the assumption in step #1 leads to a contradiction of some fact i.e. definition, postulate, corollary or theorem.
3. 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: $\angle{1}$ is an exterior angle of $\Delta{ABC}$

Prove: $m\angle{1}>m\angle{4}$

Figure 1 - Indirect Proof Diagram

Step 1 – Assume that $m\angle{1} \not> m\angle{4}$,  this means that $m\angle{1} \leq m\angle{4}$.

Step 2 – We need to establish that $m\angle{1} \leq m\angle{4}$ contradicts a mathematical fact.

$m\angle{1} \leq m\angle{4}$ gives two different situations that need to be tested:

$m \angle{1}= \angle{4}$ or $m \angle{1}.

$m \angle{1}= \angle{4}$

By the Exterior Angle Theorem, $m \angle{3}+m \angle{4}=m \angle{1}$ and using substitution, $m \angle{1}+m \angle{4}=m \angle{1}$. Subtracting $m \angle {1}$ from both sides gives $m \angle{4}=0$. This contradicts the fact an angle must have a measure greater than 0.

$m \angle{1}

By the Exterior Angle Theorem, $m \angle{3}+m \angle{4}=m \angle{1}.$ Angles must have a positive measure, the definition means $m \angle{1}> m \angle{3}$ and $m \angle{1}>m \angle {4}$.

Step 3 – In each instance, the assumption from step 1 is contradicted of a know mathematical fact. Thus, the assumption that $m\angle{1} \leq m\angle{4}$ is false. So, the original prove statement,  $m\angle{1}>m\angle{4}$ , must be true.

### 2 Responses

1. Mr. Pi,

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

Best,
Guillermo

• 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