Geometry

Here is one approach to answering the quadrilateral challenge posed earlier.  In summary, the challenge was to prove or disprove the following statement:  A quadrilateral with a pair of congruent opposite sides and a pair of congruent opposite angles is a parallelogram.

I offer this disproof without words.

By starting with an isosceles triangle, cutting it, rotating one of the pieces, and gluing it back together, we have constructed a quadrilateral with one pair of congruent opposite sides and one pair of congruent opposite angles that it is not necessarily a parallelogram!

Here’s an easy-to-understand, remarkably rich question that arose during a recent Math for America “Bring Your Own Math” workshop.

If a quadrilateral has a pair of opposite, congruent sides and a pair of opposite, congruent angles, is it a parallelogram?

I had a lot of fun thinking about this problem on my own, discussing it with colleagues, and sharing it with students.  At different times throughout the process, I felt strongly about incompatible answers to the question.  For me, that is a characteristic of a good problem.

I encourage you to play around with this.  I was surprised at how many cool ideas came out as I worked my way through this problem, and I look forward to sharing them!

And if you want to see a solution, click here.

The passing of consecutive isosceles triangle days has me once again thinking about the question “Which Triangle is More Equilateral?”

I first considered the question on 10/10/11, comparing the 10-10-11 triangle and the 10-11-11 triangle.  After a spirited discussion, I offered one approach to the question here.  The problem gave me lots to think about, both mathematically and pedagogically, and I reflected on what I liked about this problem here.

But as 12/11/11 and 12/12/11 pass, I thought I’d revisit my strategy for answering the question “Which triangle is more equilateral?”

My basic strategy, outlined in more detail here, is to ultimately to quantify the circleness of each triangle.  To me, being equilateral is all about trying to be as much like a circle as possible.  So I created a measure to determine how close to circlehood a triangle is.  Here are the numbers.

The 11-12-12 triangle’s measure is closer to 1, thus making it the more equilateral triangle.

Related Posts

• Which Triangle is More Equilateral?
• Which Triangle is More Equilateral? Part II
• Which Triangle is More Equilateral? 2012 Edition
• An Ode to Equilateralism