Use the information given in the problem to sketch a drawing of the proof. Label the knowns and unknowns. As you work through the proof, draw in necessary information that provides evidence for the proof.
Realize that a proof is just a good argument with every step justified. You can find many proofs to study online or in a textbook. [4] X Research source
Meet with your teacher out of class for extra instruction.
Being able to write a mathematical proof indicates a fundamental understanding of the problem itself and all of the concepts used in the problem. Proofs also force you to look at mathematics in a new and exciting way. Just by trying to prove something you gain knowledge and understanding even if your proof ultimately doesn’t work.
Knowing your audience allows you to write the proof in a way that they will understand given the amount of background knowledge that they have.
A two-column proof is a setup that puts givens and statements in one column and the supporting evidence next to it in a second column. They are very commonly used in geometry. An informal paragraph proof uses grammatically correct statements and fewer symbols. At higher levels, you should always use an informal proof.
For example: Angle A and angle B form a linear pair. Given. Angle ABC is straight. Definition of a straight angle. Angle ABC measures 180°. Definition of a line. Angle A + Angle B = Angle ABC. Angle addition postulate. Angle A + Angle B = 180°. Substitution. Angle A supplementary to Angle B. Definition of supplementary angles. Q. E. D.
For example: Let angle A and angle B be linear pairs. By hypothesis, angle A and angle B are supplementary. Angle A and angle B form a straight line because they are linear pairs. A straight line is defined as having an angle measure of 180°. Given the angle addition postulate, angles A and B sum together to form line ABC. Through substitution, angles A and B sum together to 180°, therefore they are supplementary angles. Q. E. D.
“If A, then B” statements mean that you must prove whenever A is true, B must also be true. [10] X Research source “A if and only if B” means that you must prove that A and B are logically equivalent. Prove both “if A, then B” and “if B, then A”. “A only if B” is equivalent to “if B then A”. When composing the proof, avoid using “I”, but use “we” instead.
For example: Prove that two angles (angle A and angle B) forming a linear pair are supplementary. Givens: angle A and angle B are a linear pair Prove: angle A is supplementary to angle B
Don’t use any variables in your proof that haven’t been defined. For example: Variables are the angle measure of angle A and measure of angle B.
Manipulate the steps from the beginning and the end to see if you can make them look like each other. Use the givens, definitions you have learned, and proofs that are similar to the one you’re working on. Ask yourself questions as you move along. “Why is this so?” and “Is there any way this can be false?” are good questions for every statement or claim. Remember to rewrite the steps in the proper order for the final proof. For example: If angle A and B are supplementary, they must sum to 180°. The two angles combine together to form line ABC. You know they make a line because of the definition of a linear pairs. Because a line is 180°, you can use substitution to prove that angle A and angle B add up to 180°.
Start by stating the assumptions you are working with. Include simple and obvious steps so a reader doesn’t have to wonder how you got from one step to another. Writing multiple drafts for your proofs is not uncommon. Keep rearranging until all of the steps are in the most logical order. For example: Start with the beginning. Angle A and angle B form a linear pair. Angle ABC is straight. Angle ABC measures 180°. Angle A + Angle B = Angle ABC. Angle A + Angle B = Angle 180°. Angle A is supplementary to Angle B.
Exceptions to using abbreviations include, e. g. (for example) and i. e. (that is), but be sure that you are using them properly. [14] X Research source
Try to apply your proof to a case where it should fail, and see whether it actually does. If it doesn’t fail, rework the proof so that it does. Many geometric proofs are written as a two-column proof, with the statement and the evidence. A formal mathematical proof for publication is written as a paragraph with proper grammar.
Q. E. D. (quod erat demonstrandum, which is Latin for “which was to be shown”). If you’re not sure if your proof is correct, just write a few sentences saying what your conclusion was and why it is significant.
