5
The parallel axiom; Euclidean
geometry
COMMENT. The effect of introducing any axiom is to narrow things down, and de-
pending on the final axiom still to be taken, we can obtain two quite distinct well-
known types of geometry. By introducing our final axiom, we confine ourselves to
the familiar school geometry, which is known asEuclidean geometry.
5.1 Theparallelaxiom...........................
5.1.1 Uniquenessofaparallelline
We saw in 4.2 that given any lineland any pointP∈lthere is at least one linem
such thatP∈mandl‖m. We now assume that there is only one such line ever.
AXIOM A 7 .Given any line l∈Λand any point P∈l, there is at most one line m
such that P∈m and l‖m.|
COMMENT.By4.2andA 7 , given any linel∈Λand any pointP∈Π, there is a
unique linemthroughPwhich is parallel tol.
Let l∈Λ,P∈Πand n∈Λbe such that l=n,P∈n and l‖n. Let A and B be
any distinct points of l and R a point of n such that R and B are on opposite sides of
AP. Then|∠APR|◦=|∠PAB|◦, so that for parallel lines alternate angles must have
equal degree-measures.
Proof.Letmbe the linePQin 4.2.1 such that|∠APQ|◦=|∠BAP|◦.Thenl‖m.
Asmandnboth containPandlis parallel to both of them, by A 7 we havem=n,so
thatR∈[P,Qand so|∠APR|◦=|∠APQ|◦. Thus|∠APR|◦=|∠APQ|◦=|∠PAB|◦.
Let l,n be distinct parallel lines, A,B∈l and P,T∈n be such that B and T are on
the one side of AP, and S=P be such that P∈[A,S]. Then the angles∠BAP,∠TPS
have equal degree-measures.
Geometry with Trigonometry
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