Recall:
   * Scissors congruence: A sc= B means figure A can be cut into pieces that can be reassembled to form figure B.
      This is also described using the word "equidecomposable".  "A and B are equdecomposable to B."

•  SC =  is a reflexive, symmetric, and transitive relation. [like congruence and similarity in geometry and equality in arithmetic]
• Theorem I : A sc= B implies Area(A) = Area(B)
• Theorem II [The converse of Theoerm I!]: Area(A) = Area(B)  implies A sc= B !!
• Simple cases as evidence and a foundation for building toward the proof of Theorem II.:
• A triangle is SC to a rectangle. Activity from last class.

 
• Problem[ "lemma"] - To find a square with the same area as a given rectangle.
  Activity from last class.

A triangle is SC to a rectangle. Activity:

• A rectangle is SC to a square.
  
  radius = r =  (a+b)/2   (#)
  
• length of DC  =  a- r
                        =  r - b
  
• x² + (CD)² =  r²
• x²  =  r² - (CD)²  [Revised! 2-3-04]
          =r² - (r-b)²
          = r² - (r² -2rb+ b²)
           =  2rb - b²
           =    (2r-b)*b     [Now notice that from (#) 2r-b = a!]
           = ab
  

• A rectangle is SC to a square.
  

 
• Two squares are SC to a single square. The Pythagorean Puzzle

 
• Triangulation .: Any polygon can be decomposed into triangles!
• A polygon is SC to a square.

 
• If two polygons have equal area, then they are SC to the same square!

Hence, any two polycons of equal area are Scissors congruent.

1. Dissection of the plane--- Tilings of the plane.
2. Activity: Tile plane with playing cards.