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

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{4}.

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}<m \angle{4}

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.

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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

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