homework

Homework assignments will be posted here every Monday, Wednesday, and Friday by 3pm. They will be due next class.

Monday, 7/22
Study for the final.
Due Wednesday, 7/24.

Friday, 7/19
Reading:
Chapter 8 of AG, stop at “Nonconvex Polygons” on pg. 166.
Exercises:
7D, 7G, 7H, 7J, 7K, 8C
Due Monday, 7/22.
In addition to the assignments above I’d also like you to send me an email letting me know what you would like to review on Monday. Anything from “I’d like to go over X homework assignment” or “I’d like to go over Y proof again” to “I’d like to practice Z, please have us do Z in class and then go over the answers”. Please send me this email by noon on Sunday.
On your final exam there will be three to four proofs chosen from the following list:
Lemma 3.3 (Ruler Sliding)
Theorem 3.39 (Opposite Ray)
Theorem 4.5 (Angle Construction)
Theorem 4.12
Theorem 5.6 (ASA)
Theorem 5.18 (Triangle Inequality)
Theorem 7.25 (Existence of Parallels)
Make sure you are prepared to prove any of the theorems from this list. There will also be one proof on the final which is not on the list above so make sure you don’t just memorize these proofs, you should practice “proofs” in general.

Wednesday, 7/17
Reading:
Chapter 7 of AG
Exercises:
7A, 7B, 7C, 7E, 7F, 7I
Due Friday, 7/19.

Monday, 7/15
Reading:
Chapter 6 of AG
Exercises:
6B, 6C, 6D, 6E, 6F
Due Wednesday, 7/17.
In class Wednesday I’m going to ask you to prove one of the following theorems for the quiz:
Theorem 3.27
Theorem 4.12
Theorem 4.28
Theorem 5.2
Trying to memorize every word in the proofs of these theorems is a bad idea, especially come the final when the list gets bigger. Instead I recommend that for each proof you try and boil down the idea into something small that you can easily memorize and reconstruct the theorem from.
For example, if you were trying to memorize a proof of ASA (Thm 5.6) you might memorize “Contradiction, assume AC > DF, cutoff AC and use the betweenness theorem on the angles at B.” That’s only one sentence, but it gives you the main idea and you should be able to fill in the details from there. One of those details, for example, is that to use the betweenness theorem (4.8) you need A * C’ * C. You get this from C’ being in the interior of the segment between A and C and you get that from the segment construction theorem. The sentence above mentions both cutting off a segment and using the betweenness theorem so you should be able to reconstruct that in between step by looking at the precise statements of the theorems you wish to use (you will have the full statement of any theorem you need available to you during the quiz/final).
After you’ve memorized the main idea try writing out the proof without peeking at the one in your book. If you consistently get stuck at one point then maybe you should memorize how to get past that point. Don’t consider yourself prepared to take the quiz until you can get through the proof of each of those theorems without peeking.
Of course, all of this is only my suggestion for how to remember proofs. If you have a good enough memory to memorize every word then feel free to do that, or if you have a third way that you’re sure works for you then feel free to do that. But on the final I plan on doing something similar to this where I give you a list of 10 or so theorems and during the test I ask you to prove two or three of my choice, so however you study make sure you can do that.
Last but not least, make sure you are still memorizing those definitions! There wont be definitions on the quiz Wednesday but there absolutely will be definitions on the final.

Friday, 7/12
Reading:
Finish Chapter 5 of AG, including the section you skipped Wednesday.
Remind yourself about interpretations and models by rereading the first page and a half of Chapter 2 of AG.
Also look over the Cartesian Plane stuff on pp. 33, 34.
Exercises:
5B, 5C, 5F, 5G, 5H, 5I
Due Monday, 7/15.
The difference between 5B and 5C is that in 5C we do not assume that the line doesn’t contain the vertices of the triangle. Also, now that the proofs are getting more complicated you should start including diagrams for all of your proofs. If a proof is extraordinarily simple then it doesn’t need a diagram, but most proofs should have a diagram; in fact, a lot of proofs should have more than one diagram. I’m going to tell the grader that he can take points off there isn’t a diagram when there should be, so to be on the safe side make sure you have diagrams for all of your proofs!

Wednesday, 7/10
Reading:
Finish Chapter 4 of AG.
Chapter 5 of AG pages 103-112 (but skip the “Intersections of Lines and Triangles” section).
Exercises:
4G, 4H, 4I, 5D, 5E
Due Friday, 7/12.
There will be a quiz friday on definitions from Chapter 4 and from the part of Chapter 5 we covered in class. Make sure you have these memorized.

Monday, 7/8
Reading:
Chapter 4 of AG, stop after Corollary 4.24 on page 97.
Exercises:
4A, 4B, 4C, 4D, 4E, 4F
Due Wednesday, 7/10.
For 4A and 4B remember to look at how 3.9 and 3.10 are proven. Your proof should be very similar. Specifically, you’ll use the fact that if you assume that two betweenness relations hold then you can get two equations like (3.2) in the proof, and this leads to a contradiction. Also a warning: for 4D take a look at Fig. 4.10 on page 93. If you assume that ∠ab and ∠cd are vertical angles then by definition ∠ab and ∠cd are proper. You don’t get to assume that ∠ad or ∠bc are proper. You will either have to prove that they are or split into cases and handle the case when they are not proper separately.

Friday, 7/5
Reading:
Finish Chapter 3 of AG.
Read pages 83 & 84 of Chapter 4 of AG.
Exercises:
3G, 3I, 3J, 3K, 3L
Due Monday, 7/8.
Also go over your proof of Theorem 2.41 (Exercise 2T). Now that we have talked more about proving an object is unique make sure you understand any objections the grader had to your proof. Try and come up with a correct proof of Theorem 2.41 using the method we discussed in class today (reread template F.13 if you need to). You don’t need to turn this in, just make sure you understand how to correctly prove Theorem 2.41.

Wednesday, 7/3
Reading:
Chapter 3 of AG, stop at “Rays” on pg. 71.
Exercises:
3B, 3C, 3D, 3E, 3F
Due Friday, 7/5.
From now on you do not need to turn in a 2-column proof. If it helps you organize your thoughts feel free to do one yourself, but you only need to turn in a paragraph style proof.

Monday, 7/1
Reading:
Appendix G of AG.
Chapter 3 of AG, stop at “Betweenness of Points” on pg. 59.
Exercises:
2O, 2Q, 2R, 2T, 2U
3A
Due Monday, 7/1.
Remember to do both a 2-column proof and a paragraph style proof for the questions from Chapter 2. For 3A you need only turn in a paragraph style proof. Also remember that in a proof you may cite any theorem in the book that comes *before* the theorem you are currently proving.

Friday, 6/28
Reading:
The rest of Chapter 2 of AG.
Appendix F of AG.
Exercises:
Do parts (a), (b), and (c) of EB and EC.
2K, 2L, 2M, 2N
Due Monday, 7/1.
Remember to do both a 2-column proof and a paragraph style proof for the questions from Chapter 2. On Monday we will go over the answers to 2J.

Wednesday, 6/26
Reading:
Chapter 2 of AG, stop at the theorems section on pg. 42.
Appendix E of AG.
Exercises:
2A, 2D, 2F, 2H, 2J (skip part m)
EA (on pp. 402)
Due Friday, 6/28.
I haven’t said this explicitly in class but make sure you really are doing the readings. There are many examples that I don’t cover and they’re discussed in much greater detail than I have time for in class. There will also occasionally be concepts that I skip over and don’t discuss in class, for example look out for the definition of “isomorphic” models in Chapter 2, you’ll need to read that before you do Exercise 2F.

Monday, 6/24
Reading:
Chapter 1 of “Axiomatic Geometry” (AG)
Book 1 of “The Elements”
For The Elements, read all the definitions, postulates, common notions, and read the statement of each theorem, you do not need to read the proof of each theorem.
Exercises:
Do 1A, 1B, 1C*, 1D from AG
*For 1C rewrite Propositions 6 and 9. Don’t do the additional discussion that the last bullet point asks you to do.
Due Wednesday, 6/26.
If you have additional time I recommend picking a proposition or two from Book I of The Elements and reading the proof carefully. Make sure you understand how each statement is supposed to follow logically from a definition, postulate, or previous proposition. Pick an early proposition with a short proof and try to prove the theorem yourself. The best way to learn how to prove things is to jump in and try. Compare your proof with Euclid’s to see if you missed any justifications.