Jacobs Geometry
Contents 6 sections

Lesson 2 — Definitions

Big idea: A definition fixes an exact meaning. The word and its definition can replace each other, so the converse of a definition is always true.

Words to know

Word Meaning
Definition The exact meaning of a word.
Converse The statement formed by swapping the hypothesis and conclusion.
If and only if (iff) Both directions are true: a → b and b → a.
↔︎ The symbol for “if and only if.”

Concepts to know

  • For a definition, the word and its definition are interchangeable.
  • So a definition gives two true conditionals:
    1. If a figure is a triangle, then it is a polygon with three sides.
    2. If a figure is a polygon with three sides, then it is a triangle.
  • The second is the converse of the first.
  • For a definition, the converse is always true.
  • Definitions are usually written with iff: > A figure is a triangle iff it is a polygon with three sides.
  • A good definition is:
    • clear and not circular;
    • not too broad;
    • not too narrow;
    • no more complicated than necessary.

Homework helper

If the homework asks… Do this…
Write a definition as two conditionals Write a → b and b → a.
Combine two conditionals Use “iff” if both directions are true.
Identify the converse Swap the hypothesis and conclusion.
Check whether a definition is good Ask: Is it circular? Too broad? Too narrow?
Decide whether a statement is a definition Ask whether the word and the statement mean exactly the same thing.
Explain why a definition’s converse is true The word and its definition have the same meaning.

Worked example

Definition: “A square is a rectangle with four equal sides.”

Two conditionals:

  1. If a figure is a square, then it is a rectangle with four equal sides.
  2. If a figure is a rectangle with four equal sides, then it is a square.

Both are true, so the definition can be written with iff.

Watch out

  • Not every true statement is a definition. “A rectangle has four sides” is true but too broad.
  • A definition cannot be circular (“a horse is a horse”).
  • Do not confuse “iff” with plain “if.” “Iff” requires both directions.
  • A definition is not proved like a theorem; it sets the meaning.

Before you turn it in