The main goal of the course will be to study the foundations of the Euclidean geometry. We will learn how to work with geometric images and how to construct careful mathematic arguments. The first part of the course will be about the basics of plane geometry: lines and points, angles, congruent triangles, quadrilaterals (parallelograms, rhombi, rectangles, etc), circles, and their properties, as well as constructions with compass and straightedge and other related topics. The second part of the course will be about "isometries" (i.e. symmetries, rotations, etc) and homotheties of the plane. We will also discuss similar figures here. Then, we will talk about plane transformations that play an important role in Lobachevskian geometry, and will make a connection to non-Euclidean geometry later in the course.
As this is an upper-level class, familiarity with proofs is expected. Indeed, one of the goals of this class is to use your MAT 200 skills to build the theory and logic of the plane geometry. Much of the material from the first part of the course will be somewhat familiar from high school, but we will take it to a very different level. You will be required to think like a mathematician, and to write careful proofs in your homework. The second part of the course will be new to most people, and somewhat more abstract.
Some sample pages of the textbook are available on the publisher's website.
(There's enough for the first week of the course; those who don't have the book yet will find it useful.)
Front Matter (Read the Foreword!)
Midterm I: Tuesday, Oct 18, in class. Midterm II: Tuesday, Nov 22, in class.
Midterm I will cover all the topics we discussed so far (see a quick checklist here; mostly this is just a list of sections from the textbook). The emphasis will be on using precise logic and proofs; you should understand the relation between direct and converse theorems, know how to work with "if and only if" statements, justify your constructions, etc. There will be 4 or 5 questions on the test. You can expect a compass-and-straightedge construction problem, a geometric locus problem, a problem based on the triangle inequality and/or use of symmetries, a parallel lines/special quadrilaterals problem, etc. Of course, basic elements such as congruent triangles will be needed as well.
The best exam preparation is to go over all past homeworks, and make sure you know how to solve similar problems. For extra practice, there's plenty of questions in the book.
You can also look at last semester's Midterm I, but please don't be mislead into thinking that there will be some "similar" problems, problems of particular "types", etc.
Midterm II: Wednesday, Nov 17, in class.
Midterm II will cover isometries and similarity (see a quick checklist here; mostly this is just a summary of the posted lecture notes). You are responsible for all previous material (you will need to use congruent triangles, parallel lines, inscribed angles, etc), but the focus will be on things covered since the first exam.
The best exam preparation is to go over all past homeworks, and make sure you know how to solve similar problems. For extra practice, you can take similarity and construction questions for the textbook; for isometries, you can just pick some compositions of rotations/translations/reflections, and ask yourself how they act and what you can say about them using our theory.
You can also look at last semester's Midterm II, but please don't expect this semester's exam to be very much like the old one.
Important: For each homework problem, please give a proof or detailed explanation as appropriate (unless otherwise stated). Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to discuss homework with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.
Week 12 (11/28 - 12/2)
Read Lecture notes on inversions Theorems 1.A and 1.B were not covered in class and are optional; the rest of material is mandatory.
Homework 12, due Tuesday, Dec 9: pdf
Week 12 (11/14 - 11/18)
Read sections 170-174, 181-183, 194-198 of the textbook; review the similarity lecture notes.
Homework 11, due Tuesday, Nov 22: pdf
Week 11 (11/7 - 11/11)
Read Lecture notes on similarity. Please refer to these as the textbook's approach is different.
Homework 10, due Nov 17: pdf
Week 10 (10/31 - 11/4)
Read sections 143-155 of the textbook; start reading the similarity lecture notes.
Homework 9, due Nov 10: pdf
Week 9 (10/24 - 10/28)
Read Lecture notes on isometries. (These notes were written by Prof. Oleg Viro in Spring 2010, with some changes and additions by Olga Plamenevskaya, 2011.)
Homework 8, due Nov 3: pdf
Week 8 (10/17 - 10/21)
Read sections 88-91 (we covered quadrilaterals before; this time, focus on their symmetries), 121, 99-101; start looking at the isometries notes (see link above).
Homework 7, due Oct 27: pdf
Week 7 (10/10 - 10/14)
Read sections 122-126, 127, 129, 132, 133. We haven't covered section 125 in class, but please read it, you'll need it for one of the homework problems (the section is quite similar to what we did discuss).
Homework 6, due Oct 20
From the textbook: 227, 231, 237, 275, 301, 306. Doing homework *before* the exam would give you some extra practice.
Week 6 (10/3 - 10/7)
Read sections 84-97, 105, 111, 112-114.
Homework 5, due Oct 13
From the textbook: 178, 179, 180, 205, 239, 245. As always, remember to prove the if-and-only-if statement in the geometric locus problem, justify your constructions in the construction problems, etc.
Weeks 4-5 (09/19 – 09/28)
Read sections 56-83.
Homework 4, due Oct 6
From the textbook: 109, 121, 124, 133, 157, 158, 166, 168. It's longer because of the break.
Week 3 (09/12 – 09/16)
Read sections 31, 44-49, 51-55.
Homework 3, due Sept 22: pdf This is a harder one, so start early! Most questions are on applications of the triangle inequality.
Week 2 (09/6 – 09/9)
Read sections 23-24, 39-43.
Homework 2, due Sep 15
From the textbook: 72, 76, 77, 79, 81, 82. Please give detailed proofs/explanations as appropriate. In particular, justify your example in 81 (i.e. explain why the corresponding elements in your triangles are congruent and why the triangles are non-congruent; do not just draw a picture).
Week 1 (08/31 – 09/02)
Read sections 1-8, 13-16, 21-22, 31-35.
We have discussed most of this in class. If don't have the book yet, use the sample pages. The reading assignments are important
because they teach you how to follow the proof
carefully, and how to build your own proofs and to use correct notation.
Homework 1, due Sep 8
In all the sections from the reading assignment, find all the instances where properties of isometries of the plane are implicitly used.
One of the properties is stated explicitly on page 2, lines 2-5: (i) One can superimpose a plane on itself or any other plane in a way that takes one given point to any other given point...
Other properties were given in the first lecture:
(ii) One can superimpose a plane on itself or any other plane in a way that takes one given ray to any other given ray.
(iii) A plane can be superimposed on itself keeping all the points of a given straight line fixed. This "flip" can be done in a unique way.
In other words,
(ii) There exists an isometry which maps a plane onto itself or any other plane in a way that takes one given ray to any other given ray.
(iii) There exists a unique non-identity isometry of a plane onto itself keeping all the points of a given straight line fixed.
Your task is to list all the places in the sections 1-8, 13-16, 21-22, 31-35 of the
textbook where these properties are used implicitly, ie
without explicit statement or reference. Present your solution as a table
page number, line number, the property, how the property is used.
(Example: p. 27, line 2, property (iii) is used to "fold" the diagram along the line BD.)
Please also do the following questions from the textbook: 2, 40, 61, 67. In all questions except 61, please give a careful explanation/proof. (To refresh MAT 200 material, read sections 28-30.) In 67, try to use a "folding argument" as in section 35.
Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.