A regular hexagon and some irregular ones will tessellate, But in general they will not.
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Yes. Tessellated hexagons are the basis of many natural structures such as honeycombs.
a
No
Yes, all triangles can be tessellated. To tessellate any triangle, take six instances of the triangle and arrange them such that they form a hexagon. This will be a repeatable pattern so being a tessellation
No.