We got two new theorems today, saw an idea for another, and then started a discussion about a thing we’ve let sit for too long.
Ms Van Donselaar took up 1.3, and proved the following theorem.
Theorem: (Van Donselaar) Let ABCD be a rhombus and a parallelogram then angle ABC is congruent to angle ADC.
Ms Pace then shared her work on 2.1, which is our first work on kites. In the end, we decided we believe only part of a generalization of the Maass-Pace-Van Donselaar Theorem 1.1.
Theorem: (Pace) Let ABCD be a kite with DA congruent to BA and BC congruent to CD. Then angle ABC is congruent to ADC.
We talked about how proper use of notation can help clarify your statements. As difficult as it can be, the reason for careful notation is to make sure things are correct and unambiguous.
The Ms Van Donselaar shared an idea for making a counterexample for the other pair of opposite angles in a kite being congruent. (Basically, stack an isosceles right triangle and an equilateral triangle.) But we need to figure out if it can actually be constructed.
Then we started a conversation about 1.5. We decided that Ms Lewis’ excellent construction only builds one kind of rhombus, but we believe there are a great many others. So, can you find conjectures or theorems about constructing other examples? Can you make “all” examples? What would that even mean?