Just as figures in a plane are made of building blocks such as points, segments,
and lines, geometric proofs are made of building blocks, too. These building
blocks include definitions, postulates, axioms, and theorems.
Together, these building blocks are combined to make each step of a proof.
In the prior two groups of Geometry SparkNotes, we
learned the definitions of many terms. These definition are used all the time
in geometric proofs. For example, a polygon
with three sides is a triangle. This is a definition. In a proof, we might be
confronted with a three-sided polygon, but not know much about it. Using the
definition of a triangle, we could deduce that the polygon is a triangle,
and with that knowledge, and our knowledge of triangles, we could deduce much
about the previously unknown polygon. In this way, definitions are used in
proofs.
Whenever a number of terms are defined, there must be a foundation of terms that
are understood to begin with. Without the use of a such terms, every definition
would be circular--words would be defined in terms of themselves. These terms
in geometry are called undefined terms. The undefined terms are the
building blocks of geometric figures, like points, lines, and planes. Points,
lines, and planes don't have specific, universal definitions. They are
explained in any text as clearly as possible, so that every following term can
be explained using a combination of undefined terms as well as previously
defined terms. Only with an understanding of these undefined terms can other
terms, like polygons, for example, be defined.
Postulates and Axioms
Postulates and axioms are statements that we accept as true without proof. They
are essentially the same thing, and in many textbooks, a distinction isn't made
between them. But more often than not, axioms are statements or properties of
real numbers, whereas postulates are statements or properties of geometric
figures.
Although axioms and postulates pertain to different concepts (numbers, and
figures, that is), they play the same role in geometric proofs: they form the
foundation of theorems. The reason we accept both without proof is that
postulates and axioms often can't be proved. Take the parallel postulate, for
example. It states that given a line and a point not on that line, exactly one
line can be drawn that contains the point and is parallel to the line. There is
no formal proof for this, but it is doubtlessly true. In this way, Postulates
and axioms are much like undefined terms. Just as undefined terms are accepted
without formal definitions (the concepts are explained, but not really defined),
and a foundation of undefined terms makes it possible to define a wide range of
other terms, postulates and axioms are understood to be true even though we have
no formal way to prove their truth. With a foundation of a few postulates or
axioms, countless theorems can be proved.
Theorems
Theorems are statements that can be proved. Once a theorem has been proved, it
can be used in other proofs. This way the bank of geometric knowledge builds up
so that one doesn't need to re-prove the simplest properties. For example, the
fact that the diagonals of a rectangle are congruent is a very basic theorem
that can be proved using the SAS method of proving triangles congruent. (The
method for formally proving this will come later.) In a future proof, you can
simply state that the diagonals of a rectangle are congruent according to this
theorem instead of having to go through the SAS method again.
Proving the truth of certain universal geometric properties is one of the
ultimate goals of geometry. Most of the things we'll prove will be congruent in
specific situations; to prove the general fact that the diagonals of any
rectangle are congruent is a much greater accomplishment.
