First, take a look at the original page on Kites and Bricks. How can these shapes be extended?

Here is the complete list of convex quadrilaterals I've looked at. The first four numbers are the squares of the lengths. Thus, on the first one, the actual lengths are 1, sqrt(2), sqrt(2), 1. The last four numbers are the tangents of the angles. 1 is a 45 degree angle, qq is infinity -- a 90 degree angle. The other angles are more irregular, but they all work with each other very nicely.

{1,2,2,1, -1,3/4,-1,4/3}, {1,1,4,4, -4/3,qq,4/3,qq}, {1,1,9,9, -3/4,qq,3/4,qq}, {1,2,1,4, -1,7,qq,4/3}, {1,2,2,9, -1,-4/3,1,4/3}, {1,2,4,9, -7,-1,3/4,qq}, {1,2,9,8, -1,7,1,-7}, {1,4,8,9, qq,-7,1,-4/3}, {1,8,2,9, -7,qq,7,qq}, {2,2,4,4, 4/3,-1,3/4,-1}, {2,2,8,8, -4/3,qq,4/3,qq}, {2,4,2,8, -1,7,qq,4/3}, {8,8,9,9, -3/4,1,-4/3,1}

The above are the pieces with only integer or sqrt[2] based sides. Parallelograms of various sorts are also possible, as well as the trapezoids below. My next thought is to find all the integer-only pentagons. All these shapes have either an integer area, or an integer area + 1/2.

{1,1,4,2, qq,qq,1,-1},
{1,2,1,8, -1,-1,1,1},
{1,2,8,9, -1,qq,1,qq},
{1,2,9,2, -1,1,1,-1},
{1,4,2,9, qq,-1,1,qq},
{1,4,9,8, qq,qq,1,-1},
{2,2,4,8, qq,-1,1,qq}

Figures by Koshi Arai

Koshi Arai -- It's not hard to make squares by selecting suitable quadrilaterals. I think it is important to set up the restriction or condition. What restriction would be good for making squares?? Send me an answer.

shapes without sqrt(5) sides. Not to scale with the above.

{1,1,5,5, qq,-2,3/4,-2},
{1,1,5,9, qq,-1/2,1/2,qq},
{1,2,2,5, -1,qq,3,2},
{1,2,4,5, 1,-1,1/2,-1/2},
{1,2,5,8, 1,-1/3,1/3,-1},
{1,4,4,5, qq,qq,2,-2},
{1,5,2,8, -1/2,3,qq,1},
{1,5,4,8, 2,-2,1,-1},
{1,5,5,5, -1/2,4/3,qq,2},
{1,5,5,9, -2,qq,2,qq},
{1,5,9,5, -2,2,2,-2},
{2,2,5,5, qq,-3,4/3,-3},
{2,4,5,5, -1,2,qq,3},
{2,4,5,5, 1,-2,3/4,-1/3},
{2,4,5,9, 1,-1/2,1/2,-1},
{2,5,5,8, -3,qq,3,qq},
{2,5,8,5, -3,3,3,-3},
{2,5,8,9, 3,-3,1,-1},
{4,4,9,5, qq,qq,2,-2},
{4,5,5,8, -1/2,4/3,-3,1},
{4,5,9,8, 2,-2,1,-1},
{5,5,8,8, -4/3,3,qq,3},
{5,5,9,9, -3/4,2,qq,2}

Tossing in sqrt(10) adds the following shapes.

{1,1,8,10,qq,-1,1/2,-3},
{1,2,5,10,-1,-3,1,3},
{1,2,9,10,1,-1,1/3,-1/3},
{1,2,10,5,-1,2,1,-2},
{1,4,5,10,qq,-2,1,-3},
{1,4,10,5,qq,3,1,-1/2},
{1,5,2,10,-2,-3,2,3},
{1,5,2,10,2,-1/3,1/2,-3},
{1,5,8,10,-1/2,3,2,3},
{1,5,10,10,-2,7,4/3,-3},
{1,5,10,10,2,-7,3/4,-1/3},
{1,8,5,10,-1,3,7,3},
{1,8,5,10,1,-3,1,-1/3},
{1,9,4,10,qq,qq,3,-3},
{2,2,10,10,qq,-2,3/4,-2},
{2,4,4,10,-1,qq,3,2},
{2,4,8,10,1,-1,1/2,-1/2},
{2,5,5,10,3,-4/3,1,-2},
{2,5,9,10,-1/3,2,3,2},
{2,5,10,5,-1/3,1,7,3},
{2,8,8,10,qq,qq,2,-2},
{2,9,5,10,-1,2,-7,2},
{2,9,5,10,1,-2,1,-1/2},
{2,10,10,10,-1/2,4/3,qq,2},
{4,4,10,10,qq,-3,4/3,-3},
{4,5,5,10,-2,qq,7,3},
{4,5,5,10,2,-3/4,1,-3},
{4,5,10,5,-1/2,1,-7,2},
{4,8,10,10,-1,2,qq,3},
{4,8,10,10,1,-2,3/4,-1/3},
{5,5,8,10,3/4,-1/3,1/2,-1},
{5,5,8,10,qq,-3,2,-7},
{5,5,10,10,4/3,-1,3/4,-1},
{5,8,9,10,-1/3,1,-3,1},
{5,9,8,10,-1/2,1,-2,1},
{5,9,10,10,2,-3,4/3,-1}

Here are more pictures by Koshi Arai.