Thanks again to Maestro Michael, making this problem a little more interesting. If you didn't see my updated code on the other thread, it looks like we have the same results up to the 20-gon. I'll have to do a timings comparison for our approaches and get back to you about that.

There's another third approach after Line + Line, and Line + Polygon, which would be Polygon + Polygon. In principle that can be done using RegionDifference and RegionIntersection, but perhaps not to tolerance. For maths purposes Region* should still be considered Experimental. I found another bad bug for RegionCongruent while testing on this data. Perhaps RegionCongruent should split out Polygons and use edge + angle code. But I haven't taken time yet to guarantee your simplified form sorting the two lists.

Thanks again Michael for "Der Zaubermeister hat über das mickey-mouse-zeug gesprochen". I'm not entirely critical of ChatGPT, but if this German grammar is incorrect, we'll have to share the blame.

Edit PS. I don't want to forget to mention these sort of "Polygon Fragmentation" algorithms, when they work correctly, are also very useful for proving theorems about difficult tilings--for example on this recent thread about David Smith's hat tile.