DEFINITION: Two figures, F and G,
are Scissors Congruent if one can be cut into polygonal
which can be rearranged to form the other.
When F is S.Congruent to G we'll write F SC= G .
I. Basic Facts about S. Congruence.
A. If F SC= G then F and G have the same area.
B. (i) F SC= F. (REFLEXIVITY)
(ii) If F SC= G then G SC= F. (SYMMETRY)
(iii) If F SC= G and G SC= H then F SC= H. (TRANSITIVITY)
By rotating the small triangle created by connecting the midpoints
two sides of a triangle 180 degrees about one of the midpoints, we
This shows that the triangle's area is the area of this parallelogram which can be computed by using the length of the base of the triangle and 1/2 of its altitude- which is the altitude of the parallelogram.
B. Any polygon can be decomposed into triangles.
THEOREM: Any two polygons F and G of equal area are S.
Proof ( and Procedure ):
1. Decompose both polygons into triangles. (III B.)
F - - S1, S2,..., Sk
G - - T1, T2,..., Tk
2. Each triangle is S. Congruent to a parallelogram. (III A.)
S1 SC= P1, S2 SC= P2 ,..., Sk SC= Pk
T1 SC= Q1, T2 SC= Q2 ,..., Tk SC= Qk
3. Each polygon is S.Congruent to a parallelogram . (II D.)
F SC= P , G SC= Q .
4. These parallelograms have equal area, hence they are S.Congruent.
P SC= Q (II C.)
5. F SC= G (I B.)