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## A Quadrilateral Challenge

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 definitely 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, this 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 some of them.

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1. December 13, 2011 at 7:03 pm

Don’t forget the angle the diagonals cross at either… that’s always another thing to factor in!

2. December 14, 2011 at 9:58 am

Ok, very nice (spoilers follow!).

If you draw the diagonal from the non-congruent angles, you it divides the quadrilateral into two triangles that are related by SSA. So then everything boils down to understanding what such pairs of triangles can look like — if any such pair of triangles is congruent, then it follows that we get only parallelogram, but if not then we can try to construct a non-parallelogram example.

3. January 7, 2012 at 3:51 pm

Great problem!! I had my students work on it the first day back after the holidays. It generated great discussions and fierce debates, lots of kids intuitively felt it “had” to be a paralellogram, but discovered they couldn’t quite prove it. It was also great in showing them how easy it is to assume things that aren’t really there. Lots of fun! Thanks!

4. January 26, 2012 at 3:25 pm

This is a very interesting problem and inspired some brilliant mathematic conversations in my classroom. Thank you!

1. December 16, 2011 at 8:55 am