Geometry with Trigonometry

Sec. 2.1 Lines, segments and half-lines 25

(iv) LetC,D∈[A,B]so thatA≤lC≤lBandA≤lD≤lB.ByA 2 (iv) eitherC≤lD
orD≤lC.
IfC≤lDandP∈[C,D],thenC≤lP≤lD. ThusA≤lC,C≤lPso by A 2 (ii),
A≤lP.AlsoP≤lD,D≤lBso by A 2 (ii)P≤lB. ThusP∈[A,B].
IfD≤lC, we interchangeCandDin the last paragraph.
(v) This follows immediately from 2.1.3.

2.1.5 Half-lines

Definition. Given a linel∈Λ, a pointA∈land a natural order≤lonl, then the set

ρ(l,A,≤l)={P∈l:A≤lP},

is called ahalf-lineorrayofl, withinitial pointA.
Given distinct pointsA,Blet≤lbe the natural order onl=ABfor whichA≤lB;
then we also use the notation[A,Bforρ(l,A,≤l).

A

B

Figure 2.4. A half-line[A,B.

A

B

C

Opposite half-lines.

As≥lis also a natural order onl,

ρ(l,A,≥l)={P∈l:A≥lP}={P∈l:P≤lA}

is also a half-line ofl, with initial pointA. We say thatρ(l,A,≤l)andρ(l,A,≥l)are
oppositehalf-lines.

Half-lines have the following properties:-

(i)In all casesρ(l,A,≤l)⊂l.

(ii)In all cases A∈ρ(l,A,≤l).

(iii)If B,C∈ρ(l,A,≤l),then[B,C]⊂ρ(l,A,≤l).
Proof.
(i) By the definition ofρ(l,A,≤l),wehaveP∈lfor allP∈ρ(l,A,≤l)and so
ρ(l,A,≤l)⊂l.
(ii) By A 2 (i)A≤lA,soA∈ρ(l,A,≤l).
(iii) AsB,C∈ρ(l,A,≤l)we haveA≤lBandA≤lC.SinceB,C∈l,byA 2 (iv)
eitherB≤lCorC≤lB.WhenB≤lC,wehaveB≤lPfor allP∈[B,C]; withA≤lB
this givesA≤lPby A 2 (ii), and soP∈ρ(l,A,≤l).WhenC≤lB, we have a similar
proof.