Back To CourseGeometry: High School
15 chapters | 160 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
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 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.
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.
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.
We know that if we prove the angle between the two sides is equal, then we can prove that the two triangles are equal by the way of the side-angle-side (SAS) postulate, which says that 'if two triangles have two sides with an angle in between that are equal, then the two triangles are congruent.' We can use the SAS postulate because it is a postulate that has been proven over the years.
So, now the question is, how can we prove the angles between the two sides in each of the triangles congruent? We see that the sides of the angles form straight lines, so we can use the vertical angles theorem to prove the angles congruent. The vertical angles theorem states that the opposite angles of two intersecting straight lines are equal to each other. Now we have a complete proof. Filled out, our two column geometry proof looks like this.
We have provided backup, or reasons, every step of the way. We have labeled our first column as statements and our second column with reasons to show that we have reasons for everything we say. One thing I want you to take note of here is that each statement is just one step closer to our hypothesis. Even if the step seems so little or so simple, you have to write it down. Nothing can be taken for granted.
Just like in a court case, the lawyer covers every angle and has a reason for everything, the same with a geometry proof. If the proof turns out to show that the hypothesis cannot be proven as true, we would write in the last line that this hypothesis cannot be proved true along with a reason.
So, what have we learned? We've learned that a geometry proof is a formal way of showing that a particular statement is true. It begins with a hypothesis and an applicable diagram. The problem might also provide some given facts you can use.
To complete the proof, you write step-by-step statements that lead to your conclusion along with reasons for each statement you make. Two columns are used to write the proof. The statements go in the first column and the reasons go in the second column.
Nothing can be taken for granted. Every little step must be written down. Every step must have a reason. If the hypothesis cannot be proven true, then you would state that at the very last line of your proof.
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Back To CourseGeometry: High School
15 chapters | 160 lessons