Rule

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

In the diagram above, the following relation holds true.

$m A B C = m A B + m B C$

Proof

Start by drawing the radii $\overline{P A},$ $\overline{P B},$ and $\overline{P C}$ and labeling the central angles.

By definition, the arc measure is equal to the measure of the related central angle. Then, $m A B = m ∠ 1$ and $m B C = m ∠ 2 .$ Finally, by the Angle Addition Postulate, it is obtained the desired result.

$m A B C = m ∠ A P C m ∠ 1 + m ∠ 2 ⇓ m A B C = m A B + m B C$