parallelogram theorems

[kclaus]

had a student ask this today and I could not give a sufficient answer.

If you know a quadrilateral has ONE pair of opposite congruent sides and ONE pair of opposite congruent angles, why isn't that sufficient to say that it's a parallelogram.

I cannot figure out how I could construct a counterexample that is not a parallelogram.

Help?

 

[pappus]

had a student ask this today and I could not give a sufficient answer.

If you know a quadrilateral has ONE pair of opposite congruent sides and ONE pair of opposite congruent angles, why isn't that sufficient to say that it's a parallelogram.

I cannot figure out how I could construct a counterexample that is not a parallelogram.

Help?

I've attached a sketch of a non-parallelogram which satisfies the given conditions:

1. The red line segments have the sam length.

2. The angles at opposite vertices have the same value.

3. A remark about the construction:

a) Start with half a parallelogram, that's a triangle.

b) Construct over the side which will be the diagonal of the paralleogram the circle which produces the the angle. In my sketch I have drawn the necessary circle with a thin black line.

c) Draw the opposite side using a compass.

 

[kclaus]

Thanks!

It's been bothering me all day!