Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro 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
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.
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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 <strong>Plane</strong> Separation Postulate says that a line has two sides; it is used todefine the concept of a half-plane.4. <strong>Plane</strong> 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
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