Plane Geometry - Bruce E. Shapiro

50 SECTION 12. VENEMA’S AXIOMS3. Ruler Postulate. For every pair of points P, Q there is a numberP Q called the distance from P to Q. For each line l there is a one-to-onemapping f : l ↦→ R such that if x = f(P ) and y = f(Q) then P Q = |x − y|.The Plane Separation Postulate says that a line has two sides; it is used todefine the concept of a half-plane.4. Plane Separation Postulate. For every line l the points that donot lie on l form two disjoint, convex non-empty sets H 1 and H 2 , calledhalf-planes, bounded by l such that if P ∈ H 1 and Q ∈ H 2 then P Qintersects l.The protractor postulate encapsulates our “common notions” (to use aEuclidean term) about angles: they can be measure, added, and ordered.5. Protractor Postulate. For every angle ∠BAC there is a realnumber µ(∠BAC) called the measure of ∠BAC such that1. 0 ≤ µ(∠BAC) < 1802. µ(∠BAC) = 0 ⇐⇒ −→ −→ AB = BC3. Angle Construction Postulate. ∀r ∈ R such that 0 < r < 180,and for every half-plane H bounded by ←→ AB, there exists a uniqueray −→ AE such that E ∈ H and µ(∠BAE) = r.4. Angle Addition Postulate. If the ray −→ AD is between the rays−→AB and −→ AC thenµ(∠BAD) + µ(∠DAC) = µ(∠BAC)As we shall see later in the discussion, the first five postulates are notsufficient to ensure that our common notions about triangle congruencewill hold (for example, the following postulate fails in taxi-cab geometry).6. SAS (Side Angle Side Postulate) If △ABC and △DEF are twotriangles such that AB ∼ = DE,BC ∼ = EF , and ∠ABC ∼ = ∠DEF then△ABC ∼ = △DEF .Parallel PostulatesThe combination of the first six postulates, when taken together, are knownas neutral geometry. They can be extended with one of three possible parallelpostulates. It turns out that the second of these – the elliptic parallelpostulate, is inconsistent with the plane separation postulate and the conceptof betweenness – but the other two postulates are each consistent withneutral geometry. In fact, it can be proven that they are mutually exclusive– if you accept either one of the Hyperbolic or Euclidean parallel postulates« CC BY-NC-ND 3.0. Revised: 18 Nov 2012