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.