Marywood University Department of Mathematics Course Information for Math 324 College Geometry by Dr. J.

. Bernhard Riemann (1826-1866) One of the developers of Elliptic Geometry.

The study of the properties and applications of axiomatic systems. It is meant to clarify and extend the main concepts of both Euclidean and Non-Euclidean geometries. The textbook is "Roads to Geometry" by Wallace and West published by Prentice Hall. The course grade is determined by 3 tests, 2 quizes, 2 Geometer's Sketchpad computer labs, a presentation of independent work, and a final examination. Sample topics of past student presentations are given below as well as important websites for the course. Although designed for math majors and minors, this course may be successfully taken by students with a solid high school background in mathematics.

Nicolai Lobachevski (1793-1856) One of the developers of Hyperbolic Geometry.

Sample Problems for Test #1

• Explain the difference between axioms and theorems. (Assume your listener is a freshman math major.)
• What does it mean for an axiom set to be consistent?
• Explain what it means for an entire axiom set to be independent.
• Show that IA 1 is independent from the other incidence axioms.
• Consider the model in which the points are the usual points in the Euclidean plane and the lines are concentric circles all having the same fixed center. Is this a model of incidence geometry? Why or why not?
• Show (by constructing a model) that the existence of parallel lines cannot be deduced from the Incidence Axioms.
• Prove the Segment-Subtraction Theorem: If A*B*C , A'*B'*C', AB @ A'B' , and AC @ A'C', then BC @ B'C'.
• Given that AC @ DF, then for any point B between A and C, there is a unique point E between D and F such that AB @ DE.
• Give two reasons why Euclid's presentation of plane geometry was flawed by today's standards.
• Prove the SSScongruence theorem.
• Given a line L and a point P not on L, prove there exists a line through P that is perpendicular to L.
• If in triangle ABC we have Angle B @ Angle C, then prove that AB @ AC.
• Suppose we take ASA as a postulate instead of SAS. Prove SAS as a theorem.
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• State and prove the Crossbar Theorem.
• State and prove Pasch's Theorem.
• Let A, B, C, D be four points, no three of which are collinear. Suppose each of the four segments AB, BC, CD, and DA does not intersect the line passing through the other two points. Prove that A is an element of Int(Angle BCD).

Sample Problems for Test #2

• List four statements that are each equivalent to Hilbert's PP.
• Lable each of the following properties as neutral, strictly Euclidean, or strictly Hyperbolic.
  • The angle sum of a triangle is always less than 180.
  • The Alternate Interior angle Theorem.
  • The opposite sides of a parallelogram are congruent.
  • If A, B, and C are points on a line, then one point must be between the other two.
  • Rectangles exist.
  • The Isosceles Triangle Theorem.
  • Two lines cannot have more than one line perpendicular to them both.
  • The Pythagorean Theorem.
  • Vertical angles are congruent.
  • Supplements of congruent angles are congruent.
  • The summit angles of a Saccheri quadrilateral are acute.
  • Given any given set of three points, there exists a unique circle containing all three of them.

Neutral Geometry

• State the Saccheri-Legendre Theorem.
• Prove the Alternate Interior Angle Theorem: If two lines are intersected by a transversal such that a pair of alternate interior angles formed is congruent, then the lines are parallel.
• Define a Saccheri quadrilateral and a Lambert quadrilateral.
• Prove the following:
  • The diagonals of a Saccheri quadrilateral are congruent.
  • The line joining the midpoints of the summit and base of a Saccheri quadrilateral is perpendicular to both.
  • The summit angles of a Saccheri quadrilateral are congruent.
  • Prove: If ABCD is a quadrilateral where angle A and angle B are right angles and angle C is congruent to angle D, then ABCD is a Saccheri quadrilateral.
  • If a Saccheri quadrilateral has both pairs of opposite sides congruent, then it is a rectangle.
  • The fourth angle of a Lambert quadrilateral cannot be obtuse.
  • If the angle sum of any triangle is a constant n, then n = 180 degrees.
• True/False:

Angle of Parallelism

• Given a line l and a point P, that is not on l, define precisely the angle of parallelism for l and P. (You will need to define at least two more points for this definition.)
• If d_o is the critical number for the point P and the line RS and if the measure of angle RPQ = d_o, then show that the ray PQ does not intersect the line RS.
• If d_o is the critical number for the point P and the line RS and if the measure of angle RPQ < d_o, then show that the ray PQ does intersect the line RS.
• Show that the angle of parallelism for a given line and point is less than 90 degrees.

 Hyperbolic Geometry

• State HyPP in terms of the angle of parallelism.
• Prove: Hilbert's PP is equivalent to the statement: The measure of the angle of parallelism is equal to 90 degrees.
• Do parallelograms exist in hyperbolic geometry? Explain.
• Are opposite sides of a parallelogram congruent? Explain.
• Prove that the following:
  • The upper base angles of a Saccheri Quad. are always acute.
  • The fourth angle of a Lambert Quad. is acute.
  • The angle sum of every right triangle is less than 180 degrees.
  • The angle sum of every triangle is less than 180 degrees.
  • Rectangles do not exist.
  • The summit of a Saccheri Quad. is longer than the base.
  • Parallel lines are not everywhere equidistant.
  • Lines with a common perpendicular cannot have a second common perpendicular.

Euclidean Geometry

• Prove the following:
  • The sum of the angles in any triangle must equal 180 degrees.
  • The measure of an exterior angle of a triangle is equal to the sum of the measures of the corresponding interior angles.
  • If a line intersects one of two parallel lines, it must intersect the other.
  • The opposite sides of a parallelogram are congruent.
  • If line l is parallel to line m and line m is parallel to line n, then l is parallel to n.
• State the Pythagorean Theorem and prove it by using the AAA Similarity Theorem.
• Suppose the circumscribed circle around a square has a radius of length one unit. What is the length of the apothem (i.e. the radius of the inscribed circle), the length of each side, the perimeter, and the area?
• Find a formula for the area of an equilateral triangle as a function of the length s of one side.
• Explain a method for finding the center of a given circle using only a compass and straight-edge.
• If seg(AB) is a diameter of a circle and if seg(CD) is another chord of the same circle that is not a diameter, then prove that AB > CD.
• Prove that the area of a regular n-gon is equal to one-half the product of its apothem and its perimeter.

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Past Student Presentations

Tesselations of the Plane (http://www.worldofescher.com) The Geometry of Surveying Isoperimetric Quadrilaterals (from Mathematics Teacher, Vol. 93, Number 2, February 2000) In Search of Perfect Triangles (from Mathematics Teacher, Vol. 92, Number 1, January 1999) Oliver Byrne's Edition of Euclid Explorations in Geometry (from Mathematics Teacher, Vol. 90, Number 5, May 1997) Kurt Godel and Incompleteness (from Scientific American) Is the Universe Finite? Cutting the Mobius Strip along the Equator Computing the Area of a Spherical Triangle Report on Flatland by A. Abbot The Moise Plane (from the College Mathematics Journal, Vol. 27, No. 3, May 1996) The History of the Pythagorean Theorem Stalking the Wild Ellipse (from the College Mathematics Journal) Models of the Klein Bottle

Useful Websites

http://www.mapleapps.com/index.asp (The Maple Applications Center)

http://www.keypress.com/sketchpad/ (Sketchpad Demos and Activities)

http://www.math.ubc.ca/~robles/hyperbolic/ (Models for Hyperbolic Geometry)

http://www.educa.fmf.uni-lj.si/java/pck/ELEMENTS/elements.html (Euclid's Elements)

http://www.math.ucf.edu/geometry/notes00f.shtml (Lecture Notes on Geometries)

http://www.sunsite.ubc.ca/DigitalMathArchive

http://xahlee.org/MathGraphicsGallery_dir/mathGraphicsGallery.html (Math Graphics)

http://math.rice.edu/~pcmi/sphere/ (Geometry on the Sphere)

http://www.maa.org (Mathematical Association of America)

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@marywood.edu

Last update: April 14, 2003

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