Mathematical Proof: Definition & Examples Video

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  • 0:03 Definitions
  • 1:09 Paragraph Proofs
  • 1:56 Flow Chart Proofs
  • 2:47 Two-Column Proofs
  • 3:15 Lesson Summary
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Lesson Transcript
Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

Have you ever made a statement that someone challenged you to prove to be true? You have to explain things in a logical and indisputable way in order to do that. In this lesson, you'll learn different ways that statements can be proven in mathematics.


Writing a mathematical proof is similar to an attorney arguing a case in a courtroom. An attorney's task is to prove a person's guilt or innocence using evidence and logical reasoning. A mathematical proof shows a statement to be true using definitions, theorems, and postulates. Just as with a court case, no assumptions can be made in a mathematical proof. Every step in the logical sequence must be proven. Mathematical proofs use deductive reasoning, where a conclusion is drawn from multiple premises. The premises in the proof are called statements.

Proofs can be direct or indirect. In a direct proof, the statements are used to prove that the conclusion is true. An indirect proof, on the other hand, is a proof by contradiction. It begins by assuming the opposite of the statement that is to be proven. During the proof, a contradiction will be reached, showing that the assumed statement is false. For the examples in this lesson, we will use direct proofs since they are used more commonly.

The format of a proof can be a simple paragraph, a flow chart, or a two-column chart. We will look at an example of each.

Paragraph Proofs

All proofs should begin with the given information that is provided. When writing a paragraph proof, each sentence provides a statement and explanation leading to the conclusion. The conclusion is the statement that is being proven.

For example:

  • Given that angle AED is a right angle, prove that angle AEC measures ninety degrees.

example of a paragraph proof

We are given that angle AED is a right angle. According to the definition of right angles, it's therefore ninety degrees. Angle AED and angle AEC are linear and supplementary, based on the definition of linear angles. If they are supplementary, then they have a sum of 180 degrees. Therefore, since AED is ninety degrees, angle AEC must also be ninety degrees.

Next, we'll see how this proof can be made using a flow chart.

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