Geometry and Trigonometry
Basics of Geometry
What are proofs and theorems in geometry?
Proofs are extremely important to geometry. Similar to other divisions of mathematics, proofs are defined as sequences of justified conclusions used to prove the validity of an “if-then” statement. (For more information about postulates, theorems, and undefined terms, see “Foundations of Mathematics.”)
There are essentially five steps in showing that any proof is a good proof: state the theorem to be proved; list what information is available; draw an illustration (if possible) to represent the information; state what is to be proved. Finally, develop a system of deductive reasoning, especially concentrating on statements that are accepted to be true; along with the true statements, add any necessary undefined terms.
In geometry, in order to prove a theorem, you need to use definitions, properties, rules, undefined terms, postulates, and (possibly) other theorems. And like hyperlinking to other text with Internet links, such theorems can be used throughout geometry (and other mathematics) in the proofs of other new, more difficult theorems.