Geometry Assignments

MTH 331: Geometry
Dr. Elmer K. Hayashi
Fall 2002
Assignments

Aug 28-30	Sep 2-6	Sep 9-13	Sep 16-20	Sep 23-27
Sep 30-Oct 4	Oct 7-11	Oct 14-18	Oct 21-25	Oct 28-Nov 1
Nov 4-8	Nov 11-15	Nov 18-22	Nov 25-26	Dec 2-6

Textbook: Wallace and West, Roads to Geometry, Second Edition

Wed, 08/28/2002. A Simple Abstract Axiomatic System: Read page 7-14.
In Sec 1.2, look at problems 1-4, and write up the solutions to problems 2 and 4 to turn in next Monday.

Fri, 08/30/2002. Properties of Axiomatic Systems.: Independence and Consistency Properties
Four Point Geometry
Four Line Geometry
Read pages 14-20.
Write up problems 15 and 24 in section 1.3 to turn in next Wednesday.
Prepare problems 5, 6, 7 in section 1.3 for discussion on Monday.

Mon, 09/02/2002. Fano's Geometry: Read pages 20-22.
On page 24, sec 1.3, Undergraduate students do problems 13, 16, and 19.
Graduate students do problems 16, 17, and 19. These problems are due on Friday.

Wed, 09/04/2002. Young's Geometry: For practice, look at problems 21 and 25 in sec 1.3 (page 24).

Fri, 09/06/2002. Euclid's Elements.: Read (scan) sec 1.4, 2.1-2.2.
Write up problems 2 and 4 in sec 2.2 (page 40) to turn in next Wednesday.

Mon, 09/09/2002. Hilbert's Axioms.: Hilbert's Axioms are carefully crafted, but somewhat tedious to apply.
Read sec 2.3-2.4.
In sec 2.4 (pages 52-53), do problems 4, 7, 8, 11 to turn in on Friday.

Wed, 09/11/2002. Birkhoff's Postulates, SMSG Postulates.: Compare and contrast Hilbert's Axioms with Birkhoff's Postulates with the SMSG Postulates.
Read sections 2.5 and 2.6.
In section 2.6 (page 64), do problem 8, and in section 3.2 (page 81), do problem 2 to turn in on Monday

Fri, 09/13/2002.: Read 3.1-3.2.
In section 3.2 (page 81), do problem 4 (line segment congruence only) and problem 5 (angle bisector only) to turn in on Wednesday.

Mon, 09/16/2002.: Redo problem 11 in sec 2.4 (page 53) to turn in Friday, i.e. prove the Crossbar Theorem.

Wed, 09/18/2002. Pasch's Axiom, Crossbar Theorem, Exterior Angle Theorem.: Read 3.2-3.3. Study proofs of 3.3.2 and 3.3.3.
Write up problem 10 in sec 3.2 (p. 81) to turn in Monday.

Fri, 09/20/2002. Triangle Inequality Theorem.: The shortest distance between two points is along the line segment joining those two points, and so the sum of the lengths of any two sides of a triangle must be longer than the third side.

Mon, 09/23/2002. Hinge Theorem.: The converse, inverse, and contrapositive of the Isosceles Triangle Theorem are all true. The Hinge Theorem follows from the Triangle Inequality Theorem, and says that if you think of two sides of a triangle as a hinge, then opening the hinge wider makes the third side of the triangle larger.

Wed, 09/25/2002. Alternate Interior Angle Theorem: It is interesting to note that the Alternate Interior Angle Theorem establishes the existence of parallel lines without assuming any statement equivalent to Pasch's Axiom (i.e. the Parallel Postulate).

Fri, 09/27/2002. First Hour Exam.: The exam will cover chapters 1-3.4.

Mon, 09/30/2002. Saccheri and Lambert Quadrilaterals.: If we assume the SAS postulate, then even without the Parallel Postulate, we can prove the ASA, AAS, and SSS theorems. The Saccheri Quadrilateral is a quadrilateral whose sides are congruent and whose base angles are right angles. It is not possible to prove that the summit angles are right angles without assuming a statement equivalent to the parallel postulate. Thus rectangles (quadrilaterals with four right angles) exist if and only if the parallel postulate is true. Without assuming the parallel postulate, it is possible to prove that the summit angles are congruent and not obtuse, and that the summit is not longer than the base.
Read sec 3.5-3.6.
Do problems 2, 3, 5 in section 3.6 (page 104) to turn in on Friday.

Wed, 10/02/2002. Saccheri-Legendre Theorem.: The angle sum of any triangle is less than or equal to 180 degrees. The parallel postulate is equivalent to the angle sum of any triangle being exactly 180 degrees.
Do problem 1 in section 3.6(pages 103-104) and prove that a Lambert Quadrilateral is essentially half of a Saccheri Quadrilateral to turn in next Monday.

Fri, 10/04/2002. The Parallel Postulate.: The converse of the Alternate Interior Angle Theorem and the angle sum of a triangle is 180 degrees are two theorems equivalent to the Parallel Postulate.
Study the list of statements equivalent to the Parallel Postulate on page 114.
In section 3.4 (page 92), do problems 5 and 6 to turn in next Wednesday.

Mon, 10/07/2002. Median Concurrency Theorem.: The existence of noncongruent similar trianges is equivalent to the parallel postulate. The medians of a triangle are concurrent at the centroid which is located two-thirds of the distance from a vertex of the triangle to the midpoint of the opposite side.
In section 4.4 (page 137), do problem 8 to turn in next Monday.

Wed, 10/09/2002. Concurrency of Perpendicular Bisectors of a Triangle.: The perpendicular bisectors of a triangle are concurrent at the circumcenter, the center of a circumscribed circle. In section 3.2 (page 81) do problem 9 and in section 4.6 (page 171) do problem 1 to turn in next Wednesday.

Fri, 10/11/2002. Concurrency of Angle Bisectors of a Triangle.: The hypotenuse-leg congruence theorem follows from the Pythagorean Theorem which in turn depends on the assumption of the parallel postulate. The angle bisectors of a triangle are concurrent at the incenter, the center of an inscribed circle.

Mon,10/14/2002.: Read about Menelaus and Ceva in section 4.7.
Do problems 2-5 in section 4.7 (pages 179-180) to check understanding.

Wed, 10/16/2002. Constructions.: Do the following constructions to turn in next Wednesday.
Construct a triangle given two sides and a median to one of the two sides.
Construct a triangle given one side, the altitude to that side, and the median to that side.
Construct a triangle given two sides, and the altitude to the third side.

Fri, 10/18/2002. Fall Break.: No Class.

Mon, 10/21/2002.: Do the following constructions to turn in next Friday.
Construct a triangle given a side, the altitude to that side and the angle opposite that side.
Construct a triangle given a side, the median to that side, and the angle opposite that side.

Wed, 10/23/2002.

Fri, 10/25/2002.: Second Hour Exam.

Mon, 10/28/2002.

Wed, 10/30/2002.

Fri, 11/01/2002.

Mon, 11/04/2002. Boundary Parallels: The Transmissibility Property of Parallelism states that if the line PX is a boundary parallel through P to the line l and R is any point of the line PX, then the PX is the boundary parallel to line l through R.
For next Friday, prove the second case of the Transmissibility Property in which R-P-X. From P drop a perpendicular to line l at Q and from R drop a perpendicular to line l at S. You must prove that if T is in the interior of angle SRX (for simplicity you may assume P-T-X), then the line RT intersects the line l.

Wed, 11/06/2002. Omega Triangles: The Crossbar Theorem and Pasch's Axiom are true for Omega Triangles
Prove the Crossbar Theorem for Omega Triangles (due next Monday).

Fri, 11/08/2002. Congruence and the Exterior Angle Theorem.: The Exterior Angle Theorem for Omega Triangles translates to an exterior angle has measure greater than the opposite interior angle. By definition two omega triangles are congruent if all three finite parts are congruent, but two prove that two omega triangles are congruent it suffices to show that two of the three parts are congruent.
Prove that two omega triangles are congruent if the two interior angles are congruent (due next Wednesday).

Mon, 11/11/2002.: The theorems on omega triangles are useful in proving the following result for Saccheri Quadrilaterals.
Prove that the Summit Angles of a Saccheri Quadrilateral are acute.

Wed, 11/13/2002.

Fri, 11/15/2002.

Mon, 11/18/2002.

Wed, 11/20/2002.

Fri, 11/22/2002.

Third Hour Exam covering constructions and Hyperbolic Geometry.

Mon, 11/25/2002.

Wed, 11/27/2002. Thanksgiving Holiday: No Class.

Mon, 12/02/2002.

Wed, 12/04/2002.

Fri, 12/06/2002.

Wed, 12/11/2002. Final Examination.: 9:00 a.m.-12:00 p.m.
Calloway 3.

Created 06/25/2002. Last modified 11/11/2002. Email to ekh@wfu.edu