The following statements will complete our discussion of areas from Friday's lecture. Statements 1–3 can be taken as axioms. There is a unique area function satisfying statements 1–3 defined on polygons, but the proof of this fact is rather complicated so we will skip it.

1. The area of a rectangle is $\text{base} \times \text{height}$.
2. Congruent figures (e.g. congruent triangles) have the same area.
3. If a figure is split into sub-figures (e.g. if a polygon is split into triangles) then the area of the figure is the sum of the areas of the sub-figures.
4. If figures $F$ and $F'$ are split into sub-figures $F_1,\ldots,F_n$ and $F_1',\ldots, F_n'$ respectively, and $F_i$ is congruent to $F_i'$ for every $i \le n$, then the figures $F$ and $F'$ have the same area. This follows from 2 and 3.
5. The area of a parallelogram is $\text{base} \times \text{height}$. This follows from 1 and 4. It is essentially the special case of Exercise 2.4.4 where one of the two parallelograms is a rectangle (see also Exercise 2.4.6.)
6. The area of a triangle is $\frac{1}{2} \text{base} \times \text{height}$. This follows from 2, 3, and 5 because to any triangle we can adjoin a congruent triangle to make a parallelogram with the same base and height (see Theorem 2.22.)