## Quadrilateral Challenge — A Solution

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!

1. Tim Ree says:

This a great exercise – because I can use it, I’m thinking, with my poetry unit. Do you have a sheet with those diagrams? Or did you just project the wordless proof on the board? Congruent angles are words, the area of triangles the form and content of the poem, and the syntax is the one variable that determines meaning… something… some parallel…

2. JBL says:

And, to tie this in to my comment on the previous thread, cutting an isosceles triangle by a cevian is exactly how you get a pair of noncongruent triangles that are related by SSA.

• MrHonner says:

Yes. I love the appearance of the Law of Sines ambiguous case here. This also offers another explanation as to why, if the opposite congruent angles are right, you can prove it’s a parallelogram.

• JBL says:

Nice; I had been wondering whether there was a “good” reason that right angles had that property (and equivalently why SSA *is* a congruence relation for right triangles); tying in the Law of Sines and the ambiguity of arcsin is a nice way to make the connection.

Somewhat further afield, this also settles the question of whether such a quadrilateral can be cyclic: if so, the opposite equal angles must be right, whence we’re stuck with just rectangles.

3. moukaouame says:

Il faut rajouter la convexité.

• MrHonner says:

My French isn’t so good: are you suggesting that we add the requirement of convexity to the quadrilateral to eliminate this counterexample?

• JBL says:

Yes, that’s the suggestion. But it doesn’t eliminate all counter-examples: you happened to choose an example that involves gluing two obtuse angles together, but it’s easy enough not to. For example, if you cut an equilateral triangle with a cevian that trisects one side and apply this operation, the resulting quadrilateral is convex.

4. MrHonner says:

Nice example. You inspired me to cut up some equilateral triangles and explore!

• Yelena Weinstein says:

Yes! That’s what I was trying to draw on the board in MfA! But it’s a cool and convincing way.

5. Derek Pipkorn says:

Showed this to my Honors Geometry class. A couple of simple questions came up: “What is the definition of an opposite side? What is the definition of an opposite angle?” This figure didn’t fit their previous knowledge of those terms.

• MrHonner says:

Yes, we had some similar conversations. What’s nice is that in a quadrilateral you can temporarily solve the problem by contrasting “opposite” with “adjacent”, and then immediately re-introduce the problem by asking the student to find the “opposite” side in a pentagon!