3 Distance; degree-measure of an angle
COMMENT. In this chapter we introduce distance as a primitive concept, relate it to
the properties of segments, and define the notion of the mid-point of two points. We
also introduce as a primitive concept the notion of the degree-measure of a wedge-
angle and of a straight-angle, relate it to the properties of interior-regions and half-
planes, and define the notion of the mid-line of an angle-support.
3.1 Distance
3.1.1 Axiomfordistance
Notation. We denote byRthe set of real numbers.
Primitive Term. There is a function||:Π×Π→Rcalleddistance. We read
|A,B|as the distance fromAtoB. We also refer to|A,B|as thelength of the segment
[A,B].
AXIOM A 4 .Distance has the following properties:-(i)|A,B|≥ 0 for all A,B∈Π;(ii) |A,B|=|B,A|for all A,B∈Π;(iii) If Q∈[P,R],then|P,Q|+|Q,R|=|P,R|;(iv) Given any k≥ 0 inR, any line l∈Λ, any point A∈l and either natural order
≤lon l, there is a unique point B∈l such that A≤lB and|A,B|=k, and a
unique point C∈l such that C≤lA and|A,C|=k.|COMMENT. Note that A 4 (iv) states that we can lay off a distancek, uniquely, on
lon either side ofA. The fact that different lettersA,B,Care used is not to be taken
as a claim thatA,B,Care distinct in all cases. Axiom A 4 (iv) implies that each linel
Geometry with Trigonometry
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