Neutral Geometry. Chapter Neutral Geometry

Transcription

1 Neutral Geometry Chapter Neutral Geometry

2 Geometry without the Parallel Postulate Undefined terms point, line, distance, half-plane, angle measure Axioms Existence Postulate (points) Incidence Postulate (lines) Ruler Postulate (distance) Plane Separations Postulate (half-plane) Protractor Postulate (angle measure) SAS Postulate (triangles combine distance and angle measure) These all replace Euclid's first four postulates Axioms for real numbers are also assumed (algebra)

3 Book 1 propositions that can be proved without the Parallel Postulate Theorems from Chapter 3 and correspondence with Euclid Existence and Uniqueness of Midpoints (Prop 10, Thm p43) Existence and Uniqueness of Angle Bisectors (Prop 9, Thm p55) Supplements and the Linear Pair Theorem (Prop 13,14, Thm 3.5.5, Def p58) Existence and Uniqueness of Perpendicular Bisectors, or Raising a Perpendicular (Prop 11, Thms 3.5.9, p60) Vertical Angles Theorem (Prop15, Thm p60) Isosceles Triangle Theorem (Prop 5, Thm p65)

4 Book 1 Propositions that can be proved without the Euclidean Parallel Postulate Theorems from Chapter 4.1 and Euclid's Propositions Exterior Angle Theorem (Prop 16, Thm p71) Existence and Uniqueness of Perpendiculars or dropping a perpendicular (Prop 12, Thm p72) Theorems from Chapter 4.2 and Euclid's Propositions ASA Congruence (Prop 26, Thm p74) Converse to the Isosceles Triangle (Prop 6, Thm p74) AAS Congruence (Prop 26, Thm p74) Copying Triangles (Prop 7, Thm p76) SSS Congruence (Prop 8, Thm p76)

5 Book 1 Propositions that can be proved without the Euclidean Parallel Postulate Theorems from Chapter 4.3 and Euclid's Propositions Scalene Inequality (Prop 18,19, Thm p77) Triangle Inequality (Prop 20, Thm p78) Hinge Theorem (Prop 24, Thm p78) Theorems from Chapter 4.4 and Euclid's Propositions Alternate Interior Angles Theorem (Prop 27, Thm p82) Corresponding Angles Theorem (Prop 28, Cor p83) Existence of Parallels (Prop 31, Cor p84)

6 The Exterior Angle Theorem - Ch 4.1 An important and fundamental theorem of Neutral Geometry The theorem is not stated in the if-then form, so starting the proof can be confusing Notice the careful setup at the beginning of the proof This is Euclid's proposition 16. Venema Chapter 1 discussed the gaps in Euclid's proof for this proposition Notice how the gaps in Euclid's proof are filled - the last part of the proof is what Euclid left out - that F is interior! The Exterior Angle Theorem does not hold on the Sphere S 2 Euclid's proof (incorrectly) works on the Sphere where triangles can have two right angles, so point F is not interior No triangle in Neutral Geometry can contain two right angles

7 Existence and Uniqueness of Perpendiculars - Ch 4.1 Thm (Existence and Uniqueness of Perpendiculars) This allows us to "drop a perpendicular" from a point to a line The use of the Plane Separation Postulate is easily overlooked The pair of cases for Q=F and Q not = F are easily overlooked Uniqueness of Perpendiculars We use the existence of perpendiculars to construct a perpendicular We use an RAA proof to show a second perpendicular can't exist We suppose a second perpendicular exists and show this leads to a contradiction with the Exterior Angle Theorem.

8 Proofs Building on SAS - Ch 4.2 Thm (ASA) Notice how the proof involves a construction of a third triangle The latter part of the proof that ray BC = Ray BC' and C =C' is often overlooked Thm (Converse to the Isosceles Triangle Theorem) This is easy to prove using ASA Let's give it a try! (Exercise 4.2.1) Thm (AAS) Let's give it a try! (Exercise 4.2.2)

9 Proofs Building on SAS - Ch 4.2 Thm (Hypotenuse-Leg) Let's give it a try! (Exercise 4.2.4) Thm (Triangle Copy) Let's give it a try! (Exercise 4.2.5) Be careful, there's an intuitive but flawed proof that implicitly relies on the 5th postulate to guarantee the point F exists! Thm (SSS) Oooh - this is a long one!

10 Triangle Inequalities - Ch 4.3 Thm (Scalene Inequality) Thm (Triangle Inquality) Thm (Hinge Theorem) Just make sure you understand the statements of these theorems

11 Distance from a Point to a Line Thm A perpendicular is the shortest distance from a point to a line. This allows us to use distance as a technical tool in proofs involving both points and lines Def introduces d(p,l) denoting distances between a point and a line as an extension to d(p,q) the distance between two points Now angle bisectors and perpendicular bisectors can be characterized by a set of points defined by an equal distance (p80).

12 Alternate Interior Angles Thm - Ch 4.4 These are the first theorems about parallel lines We prove some theorems about parallel lines in Neutral Geometry - without yet having the 5th postulate! Note the common definitions, such as a traversal (p82) to set up for theorems about parallel lines Thm (Alternate Interior Angles Theorem) is proven in the text and forms the basis for results on parallel lines Cor (Existence of Parallels) is proven in the text. Now we have demonstrated that parallel lines exist in Neutral Geometry Cor The Elliptic Parallel Postulate is false in any model for Neutral Geometry because now we know parallel lines must exist - we're left only with Euclidean and Hyperbolic Geometry