2.1: The axioms
( \newcommand{\kernel}{\mathrm{null}\,}\)
I. The Euclidean plane is a metric space with at least two points.
II. There is one and only one line, that contains any two given distinct points P and Q in the Euclidean plane.
III. Any angle AOB in the Euclidean plane defines a real number in the interval (−π,π]. This number is called angle measure of ∠AOB and denoted by ∡AOB. It satisfies the following condition:
(a) Given a half-line [OA) and α∈(−π,π], there is a unique half-line [OB), such that ∡AOB=α.
(b) For any points A,B, and C, distinct from O we have
∡AOB+∡BOC≡∡AOC.
(c) The function
∡:(A,O,B)↦∡AOB
is continuous at any triple of points (A,O,B), such that O≠A and O≠B and ∡AOB≠π.
IV. In the Euclidean plane, we have △ABC≅△A′B′C′ if and only if
A′B′=AB,A′C′=AC, and ∡C′A′B′=±∡CAB.
V. If for two triangles ABC,AB′C′ in the Euclidean plane and for k>0 we have
B′∈[AB), C′∈[AC),AB′=k⋅AB, AC′=k⋅AC,
then
B′C′=k⋅BC,∡ABC=∡AB′C′,∡ACB=∡AC′B′.
From now on, we can use no information about the Euclidean plane which does not follow from the five axioms above.
Exercise 2.1.1
Show that there are (a) an infinite set of points, (b) an infinite set of lines on the plane.
- Hint
-
By Axiom I, there are at least two points in the plane. Therefore, by Axiom II, the plane contains a line. To prove (a), it remains to note that line is an infinite set of points. To prove (b) apply in addition Axiom III.