Many "recreationally" interesting shapes may be made by tiling identical copies of the isosceles right triangle (angles of ratio 2:1:1), for example, convex polygons from 3 to 8 sides.
Similarly, the 30-60-90 right triangle (half of a equilateral triangle), the "drafter", the basis for the Eternity 1 puzzle. Angles of ratio 1:2:3.
Similarly, the pair of isosceles Robinson triangles with angles of ratios 1:2:2 and 3:1:1, which are the basis for Penrose tiles.
Similarly, the right trangle with sides 1:2:sqrt(5), half a domino, the basis for the aperiodic pinwheel tiling.
3-4-5 Pythagorean right triangle might be interesting, as all sides are commensurate.
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