Geometry with Trigonometry

56 Congruence of triangles; parallel lines Ch. 4

by 4.1.1, so by our last result,|A,D|>|A,C|.However|A,D|=|A,B|+|B,D|as
B∈[A,D], and the result follows.

P

l

m

G 1

Figure 4.9.

AB

C D

l EF

Figure 4.10.

4.3.2 Properties of parallelism

Let l∈Λbe a line, G 1 an open half-plane with edge l and P a point of G 1 .Ifmisa
line such that P∈m and l‖m, then m⊂G 1.
Proof.AsP∈l,P∈mwe havel=m.Thenasl‖mwe havel∩m=0. Thus/
there cannot be a point ofmonl. Neither can there be a pointQofminG 2 , the other
open half-plane with edgel. For then we would have[P,Q]∩l=0 and so a point/ R
ofmwould be onl,as[P,Q]⊂PQ=m.

Let AB,CD be distinct lines and l distinct from and parallel to both. If l meets
[A,C]in a point E, then l meets[B,D]in a point F.
Proof. By the Pasch property applied to[A,B,C]asldoes not meet[A,B]it meets
[B,C]at some pointG. Then by the Pasch property applied to[B,C,D],asldoes not
meet[C,D]it meets[B,D]in some pointF.

4.3.3 Dropping a perpendicular

B S A

P

l

Q R

Figure 4.11. Dropping a perpendicular.

STU

P

l

Given any line l∈Λand any point P∈l, there is a unique line m such that P∈m
and l⊥m.