Geometry

Types of Reasoning

Big Picture

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.

Key Terms

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.

Symbolic Notation

Logic uses many symbols to represent statements. Some examples of these symbols are:

Symbol
Meaning

\(\sim\)

not (negation)

\(\rightarrow\)

if-then

\(\leftrightarrow\)

if and only if

\(\therefore\)

therefore

\(\land\)

and

\(\lor\)

or

Inductive Reasoning


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.

Patterns

Examples of visual patterns:

  • Dot Patterns
  • Patterns
  • Star Patterns


Examples of number patterns:

number patterns
  • The three dots (...) means that the pattern continues.
  • Pattern: 2n, where n is the nth term in the pattern.
Examples of number patterns
  • Don’t form conjectures just by looking at the first two items in a pattern! If we look at the first two numbers, we might think the pattern is 3n. If we apply the conjecture to the rest of the numbers, we’ll see that the pattern doesn’t hold. Make sure to test any conjectures on all the items in a pattern.
  •  Pattern:  \(3 \cdot 2^{n-1}\), where n is the nth term in the pattern.

Geometry

Types of Reasoning cont.

Inductive Reasoning (cont.)

Tips for number patterns:

  • If the same number is added from one term to the next, this is the same as multiplying by that number.
  • If the same number is multiplied from one term to the next, then multiply the first term by increasing powers of this number (either n or n - 1 is the exponent).
  • If the pattern has fractions, find the patterns for the numerator and denominator separately.

Proof by Contradiction

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.

Deductive Reasoning

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.

Law of Detachment

The Law of Detachment states: If \(p \rightarrow q\) is true and p is true, then q is true.

  • If p is not true, then we cannot conclude q is true.
  • If we are given q, we cannot make a conclusion. We cannot conclude p is true.

Law of Syllogism

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.

  • This law is similar to a chain reaction or a row of dominoes falling.

Notes