Geometry: Axioms and Postulates
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.
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.