Over the course of the SparkNotes in Geometry 1 and 2, we have
already been introduced to some
postulates. In
this section we'll review
those, as well as go over some of the most important postulates for writing
proofs.

A number of postulates have to do with lines. Some are listed here.

- Through any two points, exactly one line can be drawn.
- Two lines can intersect at either zero or one
point, but no more than one.
- Through a point not on a line, exactly one line can be
drawn parallel to the first line (the parallel postulate).
- Through a point on a line,
exactly one line perpendicular to the first line can be drawn.
- Through a point not on a
line, exactly one line perpendicular to the first line can be drawn.

Other postulates have to do with measurements. Here are some.

- A segment has exactly one midpoint.
- An angle has exactly one bisector.
- The shortest distance between two points is the length of the segment
joining those points. These, though they may seem obvious, are important when
we draw auxiliary lines into figures to
write proofs.

Postulates like those in the above two lists tell us that only one
line, point, or ray of a certain type exists.

The three methods discussed for proving the congruence of triangles are all
postulates. These are the SSS,
SAS, and ASA
postulates. There is no formal way to prove that they hold true, but they are
accepted as valid methods for proving the congruence of triangles.

One final postulate has been assumed all along in the study of geometry: a given
geometric figure can be moved from one place to another without changing its
size or shape. In this text, (other than in this brief instance) we have not
and will not discuss the coordinate plane. The coordinate plane is a system in
which numbers are assigned to different locations within the plane, thus
determining the exact location of geometric figures. In this text we simply
study the figure as it exists anywhere, so it follows that it can be moved
without being changed (as far as size and shape are concerned). The postulate
simply states formally that the size and shape of a geometric figure do not
change when it is moved.

With an understanding of these postulates, as well as the
axioms discussed in the previous lessons,
we're now ready to attempt some formal proofs.