List of axioms

AXIOM A 1 .Each line is a proper non-empty subset ofΠ. For each set{A,B}of
two distinct points inΠ, there is a unique line inΛto which A and B both belong.

AXIOM A 2 .Each natural order≤lhas the properties:-

(i)A≤lA for all points A∈l;

(ii)if A≤lB and B≤lCthen A≤lC;

(iii)if A≤lB and B≤lA, then A=B;

(iv)for any points A,B∈l, either A≤lBor B≤lA.

AXIOM A 3 .Open half-planesG 1 ,G 2 with common edge l have the properties:-

(i)Π\l=G 1 ∪G 2 ;

(ii)G 1 andG 2 are both convex sets;

(iii)if P∈G 1 and Q∈G 2 ,then[P,Q]∩l= 0 /.

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.

AXIOM A 5 .Degree-measure||◦of angles has the following properties:-

(i)In all cases|α|◦≥0;

(ii)ifαis a straight-angle, then|α|◦=180;