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Rule

Arc Addition Theorem

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
A circle centered at P. Three points on it: A, B, and C.

In the diagram above, the following relation holds true.


mABC = m AB + m BC

Proof

Start by drawing the radii PA, PB, and PC, and by labeling the central angles corresponding to AB, BC, and ABC.

A circle centered at P. Three points on it: A, B, and C. Radii PC, PB, and PA. Central angles labeled.

By definition, the arc measure is equal to the measure of the related central angle. m AB &= m∠ 1 m BC &= m∠ 2 mABC&=m∠ 3 By the Angle Addition Postulate, m∠ 3 can be written as the sum of m∠ 1 and m∠ 2. mABC = m∠ 3 ⇕ mABC = m∠ 1+m∠ 2 Finally, in the above formula, m AB and m BC can be substituted for m∠ 1 and m∠ 2, respectively. mABC = m∠ 1+m∠ 2 ⇕ mABC = m AB + m BC

This theorem is sometimes accepted without a proof. For this reason, it is also known as the Arc Addition Postulate.
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