rems—another type of mathematical statement—can be proven (in addition, theo-
rems are proven by definitions or previously proven theorems). An example of a pos-
tulate in geometry is, “Through any two points there is exactly one line.” Another is,
“If two points lie in a plane, then the entire line containing those two points lies in
the plane.”What are proofsand theoremsin geometry?
Proofs are extremely important to geometry. Similar to other divisions of mathemat-
ics, 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 possi-
ble) 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 hyperlink-
ing 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.PLANE GEOMETRY
What is plane geometry?
Plane geometry is simply the study of two-dimensional figures in a plane. Most mathe-
maticians further define the plane as Euclidean (for more information about Euclid-
178 ean geometry, see above). It examines such objects as circles, lines, and polygons.
What is an indirect proof?
D
irect proofs begin with a true statement and then proceed to prove that a
conclusion is true. But there is also a method called the indirect proof in
which indirect reasoning is used. First, assume that the conclusion is false; then
show that this assumption leads to a contradiction of the hypothesis or some
other accepted fact, such as a postulate or theorem. Therefore, if the assumption
is proved false, the conclusion has been proved—indirectly—to be true.