Wentworth's Plane and Solid Geometry · Introduction, §§ 18–35 · Book 1: Introduction
2. Proofs, axioms, theorems and the general axioms
Goal: Your student knows what an axiom, theorem, postulate, problem and corollary are, can state the converse of a theorem, and knows the nine general axioms.
You'll need
- The book, open to these pages
- Pencil and the geometry notebook
What's in the book
Definitions of the general terms (proof, axiom, theorem, postulate, problem, corollary, scholium), the hypothesis and conclusion of a theorem, its contradictory, opposite and converse, then the nine General Axioms and the table of symbols.
The lesson, step by step
Warm up
Ask: How many dimensions has a surface? a line? a point? What is geometry?
The words of geometry
Read §§ 18–26 together. The four most important:
An axiom is a statement admitted to be true without proof.
A theorem is a statement to be proved.
A postulate is a construction admitted to be possible.
A corollary is a truth that is easily deduced from known truths.
If ... then
Read §§ 28–31. A theorem has a hypothesis (what is assumed) and a conclusion (what follows).
Use the book's own example in § 32: "Every horse is a quadruped" is true, but its converse, "Every quadruped is a horse," is not.
SayMake up a true sentence whose converse is false. (If it is a dog, it has four legs.)
The general axioms
Read the nine axioms on page 6 aloud. Have your student explain each with numbers, for example axiom 2: if 5 = 5 and 3 = 3, then 5 + 3 = 5 + 3.
Magnitudes which are equal to the same magnitude, or equal magnitudes, are equal to each other.
The whole is greater than any of its parts.
Symbols
Look at the table of symbols in § 35 together. Your student copies the ones for angle, triangle, perpendicular, parallel, and "therefore" into the notebook. The proofs in the book use them constantly.
Wrap up
Quick quiz: say a statement and ask, "Axiom or theorem?" Then ask for the converse of: If two angles are right angles, they are equal.
Afterwards
Learn the nine general axioms well enough to give each one by number; the proofs cite them as "Ax. 3", "Ax. 7" and so on.