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Mathematics LibreTexts

2.2: Lines and half-lines

  • Page ID
    23585
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    Proposition \(\PageIndex{1}\)

    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

    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.

    Theorem \(\PageIndex{2}\)

    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.