Geometry: Inductive and Deductive Reasoning
Building Blocks of Proof
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 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.