Polygonal Dissection Congruence Theorem 113

10.3 Parallelograms to rectangles

Lemma 77.2. Every parallelogram is dissection congruent to a rectangle.

Proof. Place the parallelogram upon one of its longest sides as a base.

→

It follows that there is a vertical from that side to its opposite (why?). By slicing on that

vertical and swapping the order of the two sides, we form a rectangle, as shown.

10.4 Rectangles to squares

Lemma 77.3. Every rectangle is dissection congruent with a square.

Proof. If we are faced with a long, thin rectangle, we can make it not quite as long and

thin by cutting it in half and stacking the two halves, like so:

→

This will double one side length and halve the other. By iterating this, if necessary, we

produce a rectangle with neither side more than twice as long as the other.

Let s be the side length of the target square we are trying to construct from the rectangle

by dissection. This is simply the square root of the area of the rectangle.

s

→ → →

Place the rectangle on its short side at the base, and construct a line of length s from the

lower left corner to the side at right, as shown. Slice on that line, cutting off the gold

triangle, and place this triangle above to form a parallelogram.