Geometric Proof

Deductive reasoning: justify each step with a postulate or theorem.

The idea

Definition (Geometric proof).

A geometric proof is a chain of statements in which every statement is justified by a stated reason: a postulate, a theorem already proved, or a property of equality.

We may not assert a claim because the picture suggests it or because it seems obvious; each line names the reason that licenses it. A two-column proof records the statements down the left column and their reasons down the right, so we can check the argument one line at a time.

Three kinds of reason are allowed. A postulate is a statement accepted as true without proof. A theorem is a statement already proved from postulates and earlier theorems; once proved, we may cite it in later proofs. A property of equality is a rule about equal quantities rather than about geometry: $a = a$ (Reflexive); if $a = b$ then $b = a$ (Symmetric); if $a = b$ and $b = c$ then $a = c$ (Transitive); if $a = b$ then $a + c = b + c$ (Addition).

Ways to work on it

• Walkthrough. Postulates versus theorems, and the properties that justify each step.
• Practice. Match a statement to its reason.
• Hardest. Find the missing reason in a short two-column proof.

Not sure where to start? Take the ten-question placement test.