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Suppose all the pentagon have two adjacent angles of 45 degrees, and three right angles. Create a line of pentagons with their bases aligned and their "odd" vertext facing upwards. Next create a second line of pentagons, inverted so as to meet the first line apex-to-apex. The gaps between these will be rectangular (square, in fact). It is thus possible to obtain a tessellation.

No tesselation is possible with regular pentagons and rectangles.

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Q: Can a pentagon and rectangle tessellate?
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