2.2: Lines and half-lines
Any two distinct lines intersect at most at one point.
- Proof
-
Assume that two lines l and m intersect at two distinct points \(P\) and \(Q\). Applying Axiom II , we get that \(l = m\).
Exercise \(\PageIndex{1}\)
Suppose \(A' \in [OA)\) and \(A' \ne O\). Show that
\([OA) = [OA').\)
- Answer
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By Axiom II , \((OA) = (OA')\). Therefore, the statement boils down to the following:
Assume \(f: \mathbb{R} \to \mathbb{R}\) is a motion of the line that sends \(0 \mapsto 0\) and one positive number to a positive number, then \(f\) is an identity map.
The latter follows from Section 1.6 .
Given \(r \ge 0\) and a half-line \([OA)\) there is a unique \(A' \in [OA)\) such that \(OA' = r\).
- Proof
-
According to definition of half-line, there is an isometry
\(f:[OA) \to [0, \infty),\)
such that \(f(O) = 0\). By the definition of isometry, \(OA' = f(A')\) for any \(A' \in [OA)\). Thus, \(OA' = r\) if and only if \(f(A') = r\).
Since isometry has to be bijective, the statement follows.