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Mathematics LibreTexts

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+BOCAOC.


(c) The function
:(A,O,B)AOB

is continuous at any triple of points (A,O,B), such that OA and OB and AOBπ.

IV. In the Euclidean plane, we have ABCABC if and only if
AB=AB,AC=AC, and CAB=±CAB.

V. If for two triangles ABC,ABC in the Euclidean plane and for k>0 we have
B[AB),          C[AC),AB=kAB,          AC=kAC,


then
BC=kBC,ABC=ABC,ACB=ACB.

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.


This page titled 2.1: The axioms is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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