We have already investigated quadrilaterals, so the logical next step is to consider pentagons. A interesting point to ponder is whether all pentagons will tessellate by themselves. We have already found techniques for any triangle or quadrilateral to tessellate by itself, so it is natural to believe that perhaps techniques for tessellating any pentagon exist.
We ask the same question as we have for quadrilaterals, triangles, and regular polygons: Is there a way to arrange a pentagon around a point to perfectly fill 360 degrees?
Recall that the sum of the interior angles of any pentagon is
540 degrees. Therefore it is impossible to fit all the angles
of pentagon into 360 degrees.
Suppose we restricted the type of pentagon allowed. Then we will be able to fit some of the pentagon's angles into 360 degrees?
If two adjacent angles of a pentagon total 180 degrees, then the pentagon will tessellate in one of two ways. Translations or reflections can be used to extend the pentagon. The following example illustrates this idea. Notice how the requirement makes it easy for the pentagon to be repeated.
In summary, any pentagon with two adjacent angles that sum to 180 degrees will tessellate in two different ways.