What proofs are made of
Two lines cross a third. Will they meet? The angles decide. Slide the two angles below.
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Step 1 of 5
- Undefined words: point, line.
- Definitions, built from words already placed.
- Axioms: the general agreements.
- Postulates: the board's own five agreements.
- Reasoning: each step forced by what came before.
Look at the two angles on one side, between the two lines. If they total under 180°, the lines meet on that side.
The layers
Five layers stack up, each on the one below. Undefined words, definitions, axioms, postulates, reasoning. Proofs use these layers. Proofs also use results already proved.
The fifth postulate at work
Two lines cross a third line. Each of the two lines makes an angle with the third. Both angles lie between the two lines. They are on the same side of the third line. Call these angles a and b.
| # | Statement | Reason |
|---|---|---|
| 1 | Angle a = 80° and angle b = 70°. | Given. |
| 2 | a + b = 150°. | Addition. |
| 3 | 150° is less than 180°. | Two right angles make 180°. |
| 4 | The lines meet on that side. | Postulate 5. |
At exactly 180°, the postulate says nothing.
Rule: A proof starts from undefined words, definitions, axioms and postulates. Each step follows by reasoning.