Lessons 14–15 · 2 lessons · Geometry Grade 7, corrected and expanded 3rd edition. Yangiyo‘l Poligraf Servis, Tashkent, 2017 (approved by the Ministry of Public Education)
14
Logical order in geometry: proofs, theorems, axioms
Textbook: pp. 36–37
GoalDistinguish proof, theorem, axiom and definition.
New words
proof · isbottheorem · teoremahypothesis and conclusion of a theorem · teoremaning sharti va xulosasiaxiom · aksioma
Explanation
Deriving the truth of a statement by logical reasoning is a proof, and a statement whose truth is proved is a theorem. A theorem has a hypothesis (what is given) and a conclusion (what must be proved): “If A holds, then B holds.” The Greek mathematician Thales (about 624–546 BC) is named as the first to bring proof into geometry. Basic notions such as point, line and plane are not defined; other notions are defined through them. Obvious properties accepted without proof are axioms; everything else is proved from axioms and already proved theorems. In a proof you may not use an unproved property other than an axiom, even if it looks obvious.
Worked examples
Theorem: “If adjacent angles are equal, each is a right angle.” Hypothesis: adjacent angles are equal; conclusion: each is 90°. Proof: α + α = 180°, so α = 90°.
“Exactly one line passes through any two points” is an axiom; “Vertical angles are equal” is a theorem (it is proved).
Class activity
“Hypothesis or conclusion?”: the teacher reads a theorem and pupils show the hypothesis with one hand and the conclusion with the other.
Practice
1
State the hypothesis of “If angles are vertical, they are equal”.
The angles are vertical
2
What is the conclusion of that theorem?
The angles are equal
3
Which is an axiom: “one line through two points” or “adjacent angles sum to 180°”?
The first
4
Why can an unproved property (other than an axiom) not be used even if it looks obvious?
Otherwise the logical structure breaks: something that looks obvious can turn out to be false.