Here you can find what has been done in the lectures, and from time to time also what is planned for upcoming lectures.
Section 24.3, 24.4 Hyperbolic Parallel Postulate. Closed triangles. Defect of a triangle. The angle sum in a triangle is less than 180 degrees. Every similarity is a congruence. There exists triangles with arbitrarily small angle sums.
Review session Plane separation postulate (Robyn), Crossbar theorem (Renee), SAS (Neel), SSS (Naoto), Saccheri quadrilateral (Peter and Andrew), Euclidean parallel postulate (Daniel).
Section 24.2 Critical parallelism.
Sections 24.1, 24.2 The all-or-none theorem. Open triangles and critically parallel rays.
Thanksgiving break No class.
Thanksgiving break No class.
Discussion of problems from Friday.
Problem session Existence of rectangles.
Handout Problems.
Section 24.1 The Critical function.
In-class Exam Covering everything in the course up to now (essentially the first 12 chapters of the book, with less emphasis on chapters 1, 8 and 9).
Section 9.2 Models for hyperbolic geometry.
The Reflection Postulate implies SAS.
Handout Postulates and Theorems. This list (without the page and date references) will be available to you during the exam.
Section 12.3 The Pythagorean Theorem.
Handout Discussion of problems from Friday
Problem session The Angle Sum Postulate, and its equivalence to the Euclidean Parallel Postulate.
Handout Problems.
Section 12.2 Similarities between triangles.
Sections 11.3, 11.4, 12.1 The comparison theorem. The basic similarity theorem. Parallel projections preserve ratios. Proportionalities.
Sections 11.1, 11.2 Trapezoids, parallelograms, and rhombuses. Parallel projections.
Section 11.1 The Euclidean Parallel Postulate. The angle sum in a triangle equals 180. Rectangles
Sections 1.8, 10.3, 10.4 Archimedean postulate for real numbers. Upper base of a Saccheri quadrilateral. The angle sum of a triangle is less than or equal to 180.
Problem session The angle sum in a triangle is less than 270. Decomposition of triangles.
Handout Problems.
Sections 10.2, 10.3 The polygonal inequality. Saccheri quadrilaterals.
Sections 7, 10.1 The Hinge theorem. The hypotenuse-leg theorem. Parallel lines. Transversals.
Fall break No class
Chapter 7 Existence of parallels. Geometric inequalities, including the triangle inequality.
Sections 6.2, 7, 6.5 Angle bisector. Exterior angles. SAA-theorem. Existence of perpendiculars.
Section 6.2 Isosceles triangles. ASA- and SSS-theorems.
Section 6.1 Discussion of what is lacking in our postulate system. Congruence of triangles. SAS Postulate.
In-class Exam Covering everything in the course up to now (essentially the first 5 chapters of the book, and our examples of models).
Handout Problems.
Problem session The dependence of the Protractor Postulate on our earlier postulates. Big protractors.
Handout Problems.
Chapter 5 Some consequences of the Protractor Postulate.
Handout Postulates and Theorems. This list (without the page and date references) will be available to you during the exam.
Chapter 5 The Protractor Postulate.
Section 4.4 Convex quadrilaterals.
Section 4.2, 4.3 Incidence theorems. The crossbar theorem.
Section 4.1, 4.2 The equivalence of the Plane Separation Postulate and Pasch's Postulate. Incidence theorems
Handout Discussion of problems from Friday
Problem session The equivalence of the Plane Separation Postulate and Pasch's Postulate.
Handout Problems
Article The Origins of Modern Axiomatics: Pasch to Peano by H. C. Kennedy
Section 4.1, 4.2 The Plane Separation Postulate. Incidence theorems.
Section 3.6, 4.1 Congruence of line segments. Convexity.
Section 3.5, 3.6 Definitions of line segments, rays, angles and triangles. Congruence of line segments.
Section 3.4 Betweenness theorems (postulates).
Section 3.4 Discussion of problems from Friday. Definition of betweenness.
Handout Discussion of problems from Friday
Problem session Distance functions and coordinate systems for different models. The triangle inequality does not follow from the ruler postulate.
Handout Problems
Section 3.3 The ruler postulate.
Handout Models of Plane Incidence Geometry
Labor Day No class
Chapter 2 Models of incidence geometry.
Chapter 2 Introduction to the course. Plane incidence postulates.
Handout Course information