Challenge Level

Here is a tessellation of regular hexagons:

Can you explain why regular hexagons tessellate?

What about a hexagon where each pair of opposite sides is parallel, and opposite sides are the same length, but different pairs of sides are not the same length?

You can print off some square dotty paper, or some isometric dotty paper, and try drawing hexagons of this form on it. You could also draw some hexagons using this interactive. Can you tessellate them?

Now let's consider hexagons with three adjacent angles which add up to $360^{\circ}$, sandwiched by two sides of equal length, as in the diagram below:

You could start by convincing yourself that sides $y$ and $z$ are parallel...

Can you find a hexagon which