Reasoning is a fundamental part of geometric proofs. There are several types of reasoning, many of which we innately and naturally picked up. However, the different types of reasoning can be categorized and are useful for understanding geometry.
Logic: The study of reasoning.
Conjecture: An educated guess based on examples in a pattern.
Inductive Reasoning: Drawing conclusions based on observations and patterns.
Counterexample: An example that disproves a conjecture.
Deductive Reasoning: Uses logic and facts to prove that relationship is always true.
Logic uses many symbols to represent statements. Some examples of these symbols are:
\(\sim\)
not (negation)
\(\rightarrow\)
if-then
\(\leftrightarrow\)
if and only if
\(\therefore\)
therefore
\(\land\)
and
\(\lor\)
or
The study of patterns and relationships is a part of mathematics. The conclusions made from looking at patterns are called conjectures. Looking for patterns and making conjectures is a part of inductive reasoning, where a rule or statement is assumed true because specific cases or examples are true.
Tips for number patterns:
We can disprove a conjecture or theory by coming up with a counterexample. Called proof by contradiction, only one counterexample is needed to disprove a conjecture or theory (no number of examples will prove a conjecture). The counterexample can be a drawing, statement, or number.
With inductive reasoning, you cannot prove the conclusion to be true. This is where deductive reasoning comes in. Deductive reasoning uses logic to prove that relationship is always true.
The Law of Detachment states: If \(p \rightarrow q\) is true and p is true, then q is true.
The Law of Syllogism states: \(\color{#04aeef}{p} \color{#000}{\rightarrow} \color{#d22027}{q}\) and \(\color{#d22027}{q} \color{#000}{\rightarrow} \color{#8bcf27}{r}\) are true, then \(\color{#04aeef}{p} \color{#000}{\rightarrow} \color{#8bcf27}{r}\) is true.