Lesson 5 — A Deductive System
Big idea: Geometry starts with undefined terms, builds definitions, accepts postulates, and proves theorems.
Words to know
| Word | Meaning |
|---|---|
| Undefined terms | Starting words left undefined: point, line, plane. |
| Postulate (axiom) | A statement accepted as true without proof. |
| Theorem | A statement proved from definitions and postulates. |
| Deductive system | The whole structure: undefined terms → definitions → postulates → theorems. |
Concepts to know
- We cannot define every word without eventually going in a circle.
- We cannot prove every statement without eventually going in a circle.
- So geometry starts with undefined terms and postulates.
- The build order is:
undefined terms → definitions → postulates → theorems- Postulate 1: Two points determine a line.
- Postulate 2: Three noncollinear points determine a plane.
- “Determine” means exactly one.
- A definition tells what a word means; a postulate is accepted; a theorem is proved.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Name the undefined terms | Point, line, plane. |
| State Postulate 1 | Two points determine a line. |
| State Postulate 2 | Three noncollinear points determine a plane. |
| Classify a statement | Definition → meaning; postulate → accepted; theorem → proved. |
| Explain why undefined terms are needed | To avoid circular definitions. |
| Explain why postulates are needed | To avoid circular proofs. |
| Use “determine” | It means exactly one, not “at least one.” |
Worked example
A tabletop supported by three duck tails illustrates Postulate 2.
- The three points where the tails touch the table are noncollinear.
- They determine exactly one plane — the tabletop.
A string hanging with a weight illustrates Postulate 1.
- Two points on the string determine the vertical line.
Watch out
- A postulate is not something you prove later; it is a starting assumption.
- “Undefined” does not mean “unknown” or “unimportant.”
- Do not confuse a definition with a theorem.
- “Determine” means exactly one line or plane.