Definition:  A quadrilateral ABCD is a parallelogram if and only if the opposite sides are parallel, i.e., AB is parallel to CD and BC is parallel to DA.

Follow this link for more about flow charts.

Properties of Parallelograms

Here is a list of statements about quadrilaterals ABCD.

1. The diagonals of ABCD bisect each other, i.e. the diagonals intersect at a point that is the midpoint of each diagonal.
2. The opposite sides of ABCD are equal.
3. Two of the sides are equal and parallel.
4. Opposite vertices of ABCD are equal.
5. Adjacent vertices of ABCD are supplementary.
6. ABCD has a point-symmetry, i.e., there is a point reflection that maps ABCD into itself. (See Brown for definition of point reflection.)

Problem 4.1 (Due, Friday, 10/24, group problem, as explained below)

Prove that the Definition and all the 6 statements A-F are equivalent.  For example, at the end, you will be able to say that if ABCD is a parallelogram, then property C holds.  You will also be able to say that if property B holds, then ABCD is a parallelogram.

• One potential way to do this, would be to prove Def if and only if A, Def if and only if B, etc. for all 6 if and only if statements.
• A second way to do this would be to prove a cycle of implications: Def implies A implies B implies C ….. implies F implies Def. 

The first method would take 12 proofs.  The second way would take 7.  In practice, you will probably need a mixture because you can’t always find a simple cycle.  If, for example, you can prove that Def if and only if C, then after that any time you have a parallelogram you always have C and any time you have C, then you have a parallelogram.

The Process and Form of Answer for 4.1

The class on Monday will break up into teams of 3 or 4 and start working on the problem then. The product will be due on Friday.  It will have two parts.

1. On the first page, will be table of contents and a “flow chart.” The table of contents will list the proofs in your document IN ORDER. (Order is critical, so that you can prove something like D implies C using B if you have already shown that D implies B before.) The flow chart will consist of letters denoting the statements and arrows showing the implications proved in your work. (Some examples showing the format of the flow chart will be on the web site so that the format will be clear.).  This front page should also have the names of all the members of your team.
2. Next, there will be proofs of each of the implications.  These should be ONE PER PAGE (the converse of a statement can be on the same page) and each one should have ONE AUTHOR who signs the page.  The page should be headed very clearly with a title such as E implies C or E if and only if F. There should be at least two implication pages, with proofs, written and signed by each member of the team.  The proofs need not be pedantic, but they should be careful and written in good style (e.g., state what is to be proved at the outset, be clear about what is given and what is being proved, give reasons for steps).