Foundations of Logical Reasoning and Proofs

To build a proof, we must first understand how we think. Inductive reasoning is the process of looking at specific examples to find a general pattern. For instance, if you see three triangles and all have internal angles summing to $180^\circ$, you might induce that all triangles do. However, patterns can be broken. Deductive reasoning, the gold standard of mathematics, uses facts, definitions, and accepted properties to reach a logically certain conclusion. If 'All $A$ are $B$' and 'This is an $A$', then 'This must be $B$'. In Geometry, we use deduction to ensure our 'blueprints' never fail.

Every geometric proof starts with a conditional statement, usually written in 'If-Then' form: $p \rightarrow q$. The 'If' part ($p$) is the hypothesis—the condition we assume to be true. The 'Then' part ($q$) is the conclusion—the result that follows. To prove a statement, we must show that whenever the hypothesis is satisfied, the conclusion is unavoidable. Be careful with the converse ($q \rightarrow p$); just because 'If it is a square, then it has four sides' is true, it doesn't mean 'If it has four sides, then it is a square' is true!

A two-column proof is a formal map of deductive reasoning. The left column contains Statements (the 'what'), and the right column contains Reasons (the 'why'). Every statement must be supported by a reason, which can be a Given (information provided), a Definition, a Postulate (an assumed truth), or a previously proven Theorem. We often use algebraic properties like the Addition Property of Equality (if $a=b$, then $a+c=b+c$) as the engine that moves our proof forward.

In geometry, we transition from numbers to shapes using specific postulates. The Segment Addition Postulate states that if point $B$ is between $A$ and $C$, then $AB + BC = AC$. Similarly, the Angle Addition Postulate allows us to add adjacent angles. We also rely on the Reflexive Property ($a = a$), which is vital when two triangles share a side. These tools allow us to bridge the gap between a visual diagram and a logical certainty.