A regular hexagon will tessellate.
Yes because it will tessellate
Some shapes tile because they fit together.
A regular octagon will not tessellate but an irregular one can.
Yes a triangle will tessellate leaving no gaps or overlaps
Sort of. Triangle is one of the shape that may tessellate(fit tightly). But one may argue that triangle is not a square (for sure), nor a tile (yet I think it may be) to a tessellate.
The rectangle is the simplest and most obvious case of a geometrical form that can tile a plane.
I would think that the obvious answer would be yes. maybe I am missing something in the question why would one think that you could not tile a seven sided polygon? * * * * * Yes, maybe you are missing something - mathematic reality. There are no polygons - regualr or irregular of 7 or more sides which will tessellate with identical shapes.
some shapes tile and some don't because if the side angles don't equal a factor of 360 degrees it will not tile and if the side angles do equal a factor a 360 degrees it willtile.
heptagon and an equilateral triangle
If it's interior angle is a factor of 360 then it will tessellate such as a square, a regular hexagon and an equilateral triangle.