2.1: The axioms

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 \((-\pi, \pi]\). This number is called angle measure of \(\angle AOB\) and denoted by \(\measuredangle AOB\). It satisfies the following condition:
(a) Given a half-line \([OA)\) and \(\alpha \in (-\pi, \pi]\), there is a unique half-line \([OB)\), such that \(\measuredangle AOB = \alpha\).
(b) For any points \(A, B\), and \(C\), distinct from \(O\) we have
\[\measuredangle AOB + \measuredangle BOC \equiv \measuredangle AOC.\]
(c) The function
\[\measuredangle: (A, O, B) \mapsto \measuredangle AOB\]
is continuous at any triple of points \((A, O, B)\), such that \(O \ne A\) and \(O \ne B\) and \(\measuredangle AOB \ne \pi\).

IV. In the Euclidean plane, we have \(\triangle ABC \cong \triangle A'B'C'\) if and only if
\[A'B' = AB, A'C' = AC, \text{ and } \measuredangle C'A'B' = \pm \measuredangle CAB.\]

V. If for two triangles \(ABC, AB'C'\) in the Euclidean plane and for \(k > 0\) we have
\[\begin{array} {rclcrcl} {B'} & \in & {[AB),} &\ \ \ \ \ \ \ \ \ \ & {C'} & \in & {[AC),} \\ {AB'} & = & {k \cdot AB,} &\ \ \ \ \ \ \ \ \ \ & {AC'} & = & {k \cdot AC,} \end{array}\]
then
\[B'C' = k \cdot BC, \measuredangle ABC = \measuredangle AB'C', \measuredangle ACB = \measuredangle AC'B'.\]

From now on, we can use no information about the Euclidean plane which does not follow from the five axioms above.

Exercise \(\PageIndex{1}\)

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.