118 Chapter 10
Become familiar with a rich collection of examples. With any mathematical topic,
explore the range of examples illustrating it. Explore diverse cases, including very
easy cases as well as complex ones, and learn how the fundamental ideas play out
with them. Know them like friends and family.
Exercises
10.1 Explain what might go wrong in the proof of lemma 77.2 if we do not place the parallelogram
on one of its longer sides.
10.2 Prove the claim in the proof of lemma 77.3 that every rectangle is dissection congruent to a
rectangle with no side more than twice as long as the other.
10.3 What is the minimum number of pieces needed to exhibit a dissection congruence between a
square and a 45-45-90 triangle?
10.4 What is the minimum number of pieces needed to exhibit a dissection congruence between a
square and a 2 ×1 rectangle?
10.5 What is the minimum number of pieces needed to exhibit a dissection congruence between a
square and two smaller squares?
10.6 Draw careful triangles of various shapes on a piece of stiff paper. Using scissors, carry out the
dissection congruence of them each with a square of the same area.
10.7 Using an ordinary piece of letter-size paper and scissors, carry out the algorithm of the text to
make it dissection congruent with a square.
10.8 True or false: Dissection congruence of polygons in the plane preserves perimeter as well as
area.
10.9 Use the figure of lemma 77.4 to give a dissection proof of the Pythagorean theorem.
10.10 There are several dissection congruence notions, depending on whether we allow arbitrary
rigid motions with the constituent pieces, or just translation, or just translation and rotation
but not reflection. Does this matter? Would the polygonal dissection congruence theorem still
hold with these stricter notions of congruence? Prove that we do not ever need to reflect in the
dissection congruence theorem—we need not ever flip the puzzle pieces over.
10.11 Prove that no disk is scissors congruent to a noncircular ellipse.
10.12 Prove that two ellipses are scissors congruent if and only if they are congruent.
10.13 Prove that a sphere and a cylinder of the same volume are not dissection congruent (allow
cutting along any plane).
10.14 Suppose you form two shapes in the plane using straight lines and arcs from a circle of radius
r. Prove that if the two shapes have the same area and used the same fragment of the circle,
then they are scissors congruent.