Plane Geometry - Bruce E. Shapiro

SECTION 12. VENEMA’S AXIOMS 51under neutral geometry, the other is provably false; and if either is takenas false, the other is provably true.Euclidean Parallel Postulate For every line l and for every externalpoint P , there is exactly one line m such that P lies on m and m ‖ lElliptic Parallel Postulate For every line l and for every external pointP , there is no line m such that P lies on m and m ‖ lHyperbolic Parallel Postulate For every line l and for every externalpoint P , there are at least two lines m and n such that P lies on bothm and m and m ‖ l and n ‖ l.Area PostulatesNeutral Area Postulate Associated with each polygonal region R thereis a nonnegative number α(R), called the area of R, such that:1. (Congruence) If two triangles are congruent, then their associate regionshave equal area; and2. (Additivity) If R = R 1 ∪ R 2 and R 1 and R 2 do not overlap, thenα(R) = α(R 1 ) + α(R 2 )Euclidean Area Postulate (Venema 9.2.2)α(R) = length(R) × width(R).If R is a rectangle, theReflectionThe Reflection Postulate (Venema 12.6.1) For every line l thereexists a transformation ρ l : P ↦→ P such that:1. If P ∈ l then ρ l (P ) = P2. If P ∉ l, then P and ρ l (P ) lie on opposite half planes of l.3. ρ l preserves distance, collinearity, and angle measure.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.