Here is a simple pentagonal shape:
Using copies of this shape it seems that you can tile the plane, without even needing to flip over the tile.
But can you really?
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Let's annotate the picture to make it easier to identify matching corners:
By vile abuse of notation let A,B,C,D,E denote the angles at the five corners. Inspecting the tiling we can read off:
360 = B+E+B+E = B+E+C+C = A+A+B = E+D+D
B+E = 180°
C = 90°
A = E+B/2 = C+E/2
D = B+E/2 = C+B/2
Checking incident edges yields:
AB = CD = AE = BC = DE
In other words the pentagon is equilateral. In particular, triangle AED is isosceles and the angle <DAE is B/2. Therefore <DAB=E, hence AD is parallel to BC (because the angles <DAB and <ABC=B sum to 180°) at which point the whole edifice comes crashing down. Indeed, it follows E=B=C=90° and A=D=135° which is incompatible with the pentagon being equilateral.