Throughout the Geometry SparkNotes, we have learned almost everything there is to know about figures in plane geometry. One of the most important ways to apply that knowledge is through writing geometric proofs. Geometric proofs are ways to logically make an argument about a certain figure or figures in geometrical terms; a formal geometric proof is a meticulous, step-by-step way to show, in most cases, that certain figures or parts thereof are congruent. Using proofs, you could measure one figure or part of a figure and know the measure of another figure or part of that figure which is impossible to measure. Naturally, just as proofs can prove congruence, they can be used to prove incongruence as well.
To learn how to write a formal proof, which is the main purpose of this text, it is necessary to backtrack a little and look at the way a mathematician must reason in order to come to useful conclusions with limited knowledge. Two of the most basic methods of mathematical reasoning are inductive and deductive reasoning. They are both useful ways to arrive at conclusions, and are both very important to the study of geometry. Deductive reasoning is used more heavily than inductive reasoning in geometry, but in all of mathematics, including some of geometry, the process of deductive reasoning is only possible after inductive reasoning has led a mathematician to hypothesize about a given situation: only after a proof has been attempted can a mathematicians hypothesis be verified or refuted.
In the following lessons, we'll see exactly how inductive and deductive reasoning are used in geometry. Finally, just as we studied the buildings blocks of geometric figures in Geometry 1, in following lessons we'll take a look at the building blocks of geometric proofs. These include definitions, postulates, axioms, and theorems. By the end of Geometry 3, we'll fully understand how to use these building blocks to write geometric proofs. The task begins in the next few lessons as we learn about the nature of mathematical reasoning.