Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”). These are the basic building blocks from which all theorems are proved (Euclid’s five postulates, Zermelo-Fraenkel axioms, Peano axioms).

A theorem is a mathematical statement that can and must be proven to be true. This means that you need to give a convincing mathematical argument as to why the line segments MUST be congruent. Here is an example of a paragraph-style proof. This is similar to a detailed explanation you might have given in the past.

Lemma: A true statement used in proving other true statements (that is, a less important theorem that is helpful in the proof of other results). Proof: The explanation of why a statement is true. • Conjecture: A statement believed to be true, but for which we have no proof.

Writing a proof consists of a few different steps. Draw the figure that illustrates what is to be proved. The figure may already be drawn for you, or you may have to draw it yourself. List the given statements, and then list the conclusion to be proved.

Flowchart proofs are organized with boxes and arrows; each “statement” is inside the box and each “reason” is underneath each box. Each statement in a proof allows another subsequent statement to be made. In flowchart proofs, this progression is shown through arrows.

Write out the beginning very carefully. Write down the definitions very explicitly, write down the things you are allowed to assume, and write it all down in careful mathematical language. Write out the end very carefully. That is, write down the thing you’re trying to prove, in careful mathematical language.

The fundamental aspects of a good proof are precision, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible. There are two different types of proofs: informal and formal.

First, a proof is an explanation which convinces other mathematicians that a statement is true. A good proof also helps them understand why it is true. Write a proof that for every integer x, if x is odd, then x + 1 is even. This is a ‘for every’ statement, so the first thing we do is write Let x be any integer.

All mathematicians in the study considered proofs valuable for students because they offer students new methods, important concepts and exercise in logical reasoning needed in problem solving. The study shows that some mathematicians consider proving and problem solving almost as the same kind of activities.