# 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 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.

## 4 Comments

## Colin · 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!

## JBL · 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.

## Tom Baker · 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!

## Robyn · January 26, 2012 at 3:25 pm

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