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2.2: Lines and half-lines

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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

$$[OA) = [OA').$$

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.