This is the first in a potentially long series of solitaire games and puzzles that you can make yourself. Below you see two pairs of quadrilaterals, each pair consisting of mirror symmetric copies. Print & cut them out, you will need two sets to solve the puzzles.
These puzzle pieces consist of two right triangles that are glued together along their respective hypothenuse. I am using here triangles for which the short legs have lengths 1 and 8 for the first triangle and 4 and 7 for the other. Turning these triangles over gives the four possibilities above.
There are various ways these triangle fit together so that vertices meet vertices, and this creates many different ways to tile the plane.
My choice of edge lengths is important, because 1²+8²=4²+7². This not only makes the hypothenuses of the two triangles equal, but also allows to fit the quadrilaterals together so that a vertex of one triangle meets another triangle along one of its edges, as the top figure shows.
I don’t know whether there are such vertex-edge tilings of the entire plane, periodic or not.
Below are three puzzles. Use two sets of the four quadrilaterals to tile the shapes below without overlaps or gaps. I hope they are not too easy.
The Inner Frame 