The decomposition of polygons
A. T. Olson · International Journal of Mathematical Education in Science and Technology · 1983
In this paper a number of interesting results are defined and explained that are related to the notions of piecewise congruence by addition or subtraction. Two polygons are piecewise congruent by addition if either polygon can be cut into a finite number of polygonal pieces that can be rearranged to cover the other polygon. Initially, the fundamental theorem, that two polygons equal in area are also piecewise congruent by addition, is treated. Then it is shown, additionally, that there always exists a decomposition in which the sides of corresponding parts are parallel. Furthermore, for certain pairs of polygons, decompositions can be found which require only translations in relating corresponding parts. The necessary conditions for this theorem are found by using the boundary descriptions of the polygons under consideration.