Working with parallelograms

Today we finished working through the Parallelogram Conjectures and Proofs problems and a few additional problems pulled from an old textbook. Students had actually worked through many of the problems at home and seemed to be understanding what they were doing.

On the new set of problems, students were recognizing that because they had a parallelogram that opposite sides were congruent and, therefore, the expressions denoting side lengths had to be equal. The same went with opposite angles. I was pleased to see this recognition happening.

We did go through answers for the Parallelogram Conjectures and Proofs problems in class. The last set of questions related to notation went well, although there were a couple of minor notation issues that had to be addressed. Problem 10 on the second page was an issue for students.

 Students were able to find values for x and y but not z. I had a couple of minutes to play around with it in front of class and started to suspect that there wasn't enough information to solve for z. I asked students to play around with different relationships over the weekend to see if they could come up with anything.

Over lunch, I set up missing angles as variables and established a 4 x 4 matrix to represent the system of equations. The matrix is singular, so the system I set up does not have a solution.

This will be a good point of discussion. At what point do you start to suspect that there isn't a solution or the statement is not true. I can pull in a couple of historical references (5th postulate or general solution for the quintic) to tie into this point.