Introduction to Proof: Reasoning in Geometry

Language of Reasoning

If-then diagram showing that the output depends on the input.

Just like any language, the reasoning and logical processes have their own symbols and rules. These have been developed to represent the reasoning processes by a common language that makes sense to everyone who knows and understands mathematical proofs. Here are some of the basic symbols and terms we use in mathematical reasoning:

Statements and their Values

Any expression that declares a fact or an idea is called a simple statement. A simple statement is either true or false.


For example, in the following list of expressions, (2), (3), and (4) are simple statements.

  1. Why is ABC an equilateral triangle?
  2. The sum of 21 and 143 is greater than the difference of 345 and 223.
  3. The sum of angles of a square is 360 degrees.
  4. The sum of angles of a rectangle is greater than 360 degrees.
  5. Why is a square different from a rectangle?

Statements (1) and (5) do not express any facts, rather they ask about some information. Among the statements (2 - 4), statement (4) is false and the others are true.

Simple statements are denoted by p, q, r, s, and t, and usually are called just "statements." Therefore, when we talk about statements we mean simple statements.

If a statement is true, its value is denoted by T and if it is false, its value is denoted by F.