Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
154 SECTION 31. QUADRILATERALS IN NEUTRAL GEOMETRYProof. Since □ABCD is a convex quadrilateral then σ(□ABCD) ≤ 360.Since the sum of the first three angles is 270, the remaining one must be≤ 90.Corollary 31.25 Let □ABCD be a Lambert quadrilateral with right anglesat vertices A, B, and C. Then BC ≤ AD.Figure 31.9: In a Lambert quadrilateral, BC ≤ ADProof. Suppose BC > AD (RAA).Then there exists a point E with B ∗ E ∗ C such that BE = AD (rulerpostulate).□ABED is a Saccheri quadrilateral (def. of Saccheri quadrilateral).Hence ∠BED ≤ 90 (theorem 31.20).Angle ∠BED is an exterior angle for △ECD.Angle ∠C = 90, and it is a remote angle of ∠BED, By the exterior angletheorem (theorem 24.4), ∠C < ∠BED (strict inequality). This leads to theresult 90 < 90; therefore we must reject the RAA hypothesis and concludethat BC ≤ AD.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 32The Euclidean ParallelPostulateIn this section we consider some statements that are equivalent to Euclid’s5th postulate.When we say that two statements are equivalent in this sense we meanthat if we add either statement to the axioms of Neutral Geometry, we canprove the other statement. It does not mean that the two statements areprecisely logically equivalent.Axiom 32.1 .Euclidean Parallel Postulate (EPP). For every line l andfor every point P that does not lie on l there is exactly one line m suchthat p ∈ m and m ‖ l.Axiom 32.2 Euclid’s Fifth Postulate (E5P). Let l and m be two linescut by a transversal in such a way that the sum of the measures of the twointerior angles on one side of t is less than 180. Then l and m intersect onthat side of t.Theorem 32.3 Euclid’s 5th postulate ⇐⇒ the Euclidean Parallel Postulate.Proof. (⇒) (EPP ⇒ E5P). Assume that the EPP is true.Let l, m, t, α, β be as indicated in figure 32.1, i.e., construct the lines l,m, and n as shown, with then α + β < 180. We need to show that lines land m intersect on the same side of t as α and β.There is a line n through B such that γ + δ = 180 (by the protractor155
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Section 32The Euclidean ParallelPostulateIn this section we consider some statements that are equivalent to Euclid’s5th postulate.When we say that two statements are equivalent in this sense we meanthat if we add either statement to the axioms of Neutral <strong>Geometry</strong>, we canprove the other statement. It does not mean that the two statements areprecisely logically equivalent.Axiom 32.1 .Euclidean Parallel Postulate (EPP). For every line l andfor every point P that does not lie on l there is exactly one line m suchthat p ∈ m and m ‖ l.Axiom 32.2 Euclid’s Fifth Postulate (E5P). Let l and m be two linescut by a transversal in such a way that the sum of the measures of the twointerior angles on one side of t is less than 180. Then l and m intersect onthat side of t.Theorem 32.3 Euclid’s 5th postulate ⇐⇒ the Euclidean Parallel Postulate.Proof. (⇒) (EPP ⇒ E5P). Assume that the EPP is true.Let l, m, t, α, β be as indicated in figure 32.1, i.e., construct the lines l,m, and n as shown, with then α + β < 180. We need to show that lines land m intersect on the same side of t as α and β.There is a line n through B such that γ + δ = 180 (by the protractor155