# Math 362: Survey of Geometry

### Title

Survey of Geometry.

(3:3:0)

F, W, Su

### Description

This course studies the foundations of geometry going back more than two thousand years to Euclid and the ancient Greeks. This course is especially aimed at understanding the importance of Euclid’s parallel postulate and the alternative non-Euclidean geometries that arise from alternative axioms. This course places an emphasis on logical thinking and clear mathematical writing. Geometry software such as Geometer’s Sketchpad should be used throughout the course when appropriate.

## Desired Learning Outcomes

Students should gain familiarity with axioms of geometry, both Euclidean and non-Euclidean. Students should be able to prove the major theorems of geometry based on the axioms.

### Prerequisites

A knowledge of calculus and a maturity developed in mathematical communication.

### Minimal learning outcomes

Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove many of the theorems.

1. Euclid’s Elements
• Understand the historical importance of Euclid’s Elements.
• Understand and interpret Euclid’s definitions, axioms, and common notions.
• Identify the logical gaps in Euclid’s proofs.
2. Axiomatic Systems and Incidence Geometry
• Give examples of axiomatic systems and understand the axioms for Incidence Geometry.
• Give examples of systems that satisfy the axioms of Incidence Geometry.
• State theorems about Incidence Geometry.
• Write direct and indirect proofs of theorems of Incidence Geometry.
3. Set Theory and Real Numbers
• Understand basic facts from set theory including sets, elements, intersections, unions, and distinguish between “element of” and “subset of.”
• Write correct mathematical statements using the vocabulary of set theory.
• Understand the properties of the real numbers including trichotomy, density of rational numbers, the Archimedean property, and the least upper bound property.
4. Axioms for Plane Geometry
• Identify undefined terms for neutral geometry.
• Understand existence and incidence postulates and vocabulary for neutral geometry in terms of basic facts from set theory.
• Give examples of different metrics to show an understanding of the Ruler Postulate and coordinate functions on lines.
• Understand the need for and use the Plane Separation Postulate, and use it to define the interior of polygon.
• Define angles, and angle measure using the Protractor Postulate.
• Define and prove theorems about betweenness for points and rays.
• Prove theorems about vertical angles, linear pairs, and perpendicular bisectors.
• Define congruence of triangles and the need for the side-angle-side Postulate.
• Understand the Euclidean, elliptic, and hyperbolic parallel postulates and examples of geometries that satisfy each of the parallel postulates.
5. Theorems in Neutral Geometry
• Prove theorems about isosceles triangles and perpendicular lines.
• Prove and understand the Exterior Angle Theorem.
• Prove and understand congruence theorems for triangles.
• Prove and understand triangle inequality theorems relating sides and angles (Scalene Inequality, Triangle Inequality, Hinge Theorem).
• Prove and understand theorems about parallel lines cut by a transversal.
• Prove and understand the Saccheri-Legendre Theorem about the sum of the angles of a triangle.
• Be able list and prove that statements are equivalent to the parallel postulate.
6. Basic Theorems of Euclidean Geometry
• Prove and understand the Euclidean theorems about parallel lines cut by a transversal.
• Prove and understand the angle sum theorem for triangles.
• Prove and understand theorems about quadrilaterals including squares, rectangles, parallelograms, and trapezoids.
• Prove and understand theorems about the ratio of lengths of segments of transversals to three parallel lines.
• Prove and understand theorems about similar triangles.
• Prove and understand the Pythagorean Theorem.
• Work with triangles including altitudes, medians, angle bisectors and perpendicular bisectors. Prove and understand the concurrency theorems and the Euler Line Theorem.
7. Hyperbolic Geometry
• Prove and understand the angle sum theorem for triangles and define the angle defect of a triangle.
• Prove and understand theorems about parallel lines and transversals.
• Prove and understand that triangles with congruent angles are congruent.
• Describe and understand limiting parallel rays and asymptotically parallel lines.
• Model the hyperbolic plane with the Poincaré disk or the upper half plane.
8. Transformations
• Contrast the transformational perspective with the Euclidean perspective of Euclid.
• Define an isometry and prove that isometries for a geometry from a group.
• Show that translations, rotations, and reflections are isometries.
• Prove that the group of isometries for the plane is generated by reflections about a line.
• Develop plane geometry by replacing the side-angle-side postulate by the reflection postulate and defining congruence in terms of isometries.
9. Van Hiele Levels
• Be familiar with the van Hiele Model of the development of geometric thought.

### Textbooks

Possible textbooks for this course include (but are not limited to):