The logic of planimetry and problem-solving methods
Lessons 16–17 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
16
The logical structure of planimetry
Textbook, Part 1: pp. 97–101
GoalKnow the axiomatic structure of geometry and the notions of axiom, theorem and definition; separate the hypothesis and conclusion of a theorem; know the history of Euclid’s fifth postulate.
“Geometry” is Greek for “earth measurement”; planimetry studies figures in a plane and stereometry figures in space. Geometry is built axiomatically: first basic figures such as the point and the line are accepted without definition, then their basic properties — the axioms — are accepted without proof, and all other figures are defined and their properties are proved as theorems. The axioms of planimetry fall into five groups: incidence, order, measurement, placing equal figures, and the parallel axiom. A theorem has a hypothesis (what is given) and a conclusion (what must be proved); a proof may use only axioms and previously proved theorems, even an “obvious” but unproved fact is not allowed. Euclid (around the beginning of the 3rd century BC) set out geometry in his Elements in 13 books with five postulates; the fifth postulate (about parallel lines) looked more complicated than the others, so for centuries people tried to prove it, among them Omar Khayyam. N. I. Lobachevsky (1792–1856) showed that it does not depend on the other axioms and created non-Euclidean geometry (where the angle sum of a triangle is less than 180°); Ya. Bolyai and K. Gauss contributed to this too.
Worked examples
Classification: “Exactly one line passes through two different points” — axiom; “Vertical angles are equal” — theorem; “A segment is the part of a line between two points” — definition.
Theorem: “If two angles are vertical, they are equal.” Hypothesis — two vertical angles ∠1 and ∠3; conclusion — ∠1 = ∠3. Proof: ∠2 is adjacent to each of them, so ∠1 + ∠2 = 180° and ∠3 + ∠2 = 180°; hence ∠1 = ∠3.
Class activity
“Axiom or theorem?” game: the teacher reads geometric statements; students raise a hand and answer “axiom”, “theorem” or “definition” and explain why.
Practice
1
State the three stages of the axiomatic construction of geometry.
1) basic figures are accepted without definition; 2) their basic properties are accepted without proof (axioms); 3) other figures are defined and their properties proved.
2
Separate the hypothesis and conclusion of the theorem “if a triangle is isosceles, its base angles are equal”.
Hypothesis: the triangle is isosceles. Conclusion: its base angles are equal.
3
Two adjacent angles sum to 180° (measurement axiom). If one is 58°, how many degrees is the other?
122
4
Why can an “obvious” but unproved fact not be used in a proof?
A proof must rest only on axioms and proved theorems. An “obvious” fact may even be false or independent: Euclid’s fifth postulate also looked natural, yet it could not be derived from the other axioms.