Geometric Proofs for Polygons

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  • 0:02 What Is a Geometry Proof?
  • 1:01 Why Important?
  • 2:11 Hypothesis
  • 2:50 Two Column Proof
  • 5:41 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn why it is important for you to learn how to do formal proofs in geometry. Also learn how to set up a proof and how to successfully execute one.

What Is a Geometry Proof?

A geometry proof is a formal way of showing that a particular statement is true. It uses a systematic method of showing step-by-step how a certain conclusion is reached. It provides full backup every step of the way.

Every geometry proof begins with a hypothesis, or statement that may or may not be true, along with a diagram if applicable. The proof begins by writing down everything that is known to be true about the situation. Then, it proceeds to show step-by-step how the hypothesis is either true or false by providing reasons based on the given facts or other known facts.

You can liken a geometry proof to a court case where the judge needs to see fully executed proof before giving his judgment. A fully executed proof would be the successful case that a lawyer builds with evidence showing how he reached his conclusions.

Why Important?

Why are geometry proofs important to learn? Because once you have learned how to write one, you can show the world how the answer or conclusion that you have reached is true. If anyone raises a question as to whether your answer is really correct, you can show the person your proof. The proof shows that you know what you are talking about and it provides backup to answer any other questions.

If mathematicians didn't write geometry proofs, we wouldn't have all the theorems we have now. The system we have now of proving new statements about geometry by way of proofs is actually the basis of geometry. Because of this system of proofs, we can say for certain that, for example, the Pythagorean theorem is true and can be used for all right triangles.

So, yes, learning how to do geometry proofs is important and is a necessary skill for the advancement of math. Now that we've covered what a geometry proof is and why it is important, let's see the components of a geometry proof in more detail.

Hypothesis

The first part of a geometry proof is the hypothesis with a diagram, if it fits. Along with this hypothesis, some facts may also be provided about the situation. For example, the problem might ask us to prove the hypothesis 'triangle ABC is congruent to triangle EDC.' The problem also provides some given facts, such as line segment AC is congruent to line segment EC and line segment BC is congruent to line segment DC. Properly set up, the problem will look like this.

Two Column Proof

The actual proof begins after the problem is presented. The best way to present a geometry proof is with the use of a two column system. The first column is used to present the step-by-step thoughts that lead up to the conclusion, while the second column provides the backup every step of the way.

Backup for geometry proofs can be the given facts of the problem itself, as well as other proven theorems and postulates that you have learned in your geometry class. Math has been around for thousands of years and by doing just a bit of research, you can find that there are many theorems and postulates that have been proven over the years. You can use any one of these as a backup.

Going back to our problem, let's see how this proof is worked. We want to prove the hypothesis that the two triangles are congruent. We are given the facts that two of the sides of the triangles are equal to each other. We have marked our diagrams with that information already. We just need to think of what we need in order to prove our hypothesis.

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