Theorems on tessellations by polygons

What general regularity manifests itself in the fact that a triangle, and in general any convex polygon, cannot be tessellated by non-convex quadrangles? Another question: it is known that for $n>6$ the plane cannot be tessellated by convex $n$-gons if their diameters are bounded, while the areas are separated from zero; can this fact be generalized for non-convex polygons? In the present paper we introduce the characteristic $\chi(M)$ of a polygon $M$. We answer the above questions in terms of $\chi(M)$ and then study tessellations of the plane by $n$-gons equivalent to $M$, that is, with the same sequence of angles greater than and smaller than $\pi$.