**Theorem 5.4**: (The angle addition axiom) Let *C *be a
point on the arc between *A *and *B*. Then the length of the arc from *A *to *B *is the sum of the lengths of the arcs from *A *to *C *and from *C *to *B*.

**Proof**: If *C *is not in a partition, we will get a
better approxmation by adding it to the
partition. In all of the
partitions which include *C *the
lengths of the inside and
outside
approximations of the arc
length from *A *to *B *will be equal to the lengths of the inside and
outside
approximations of the arc
length from *A *to *C *plus the lengths of the inside and
outside
approximations of the arc
length from *C *to *B*.