2.2: Lines and half-lines

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.