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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.

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 by labeling the central angles corresponding to $A B,$ $B C,$ and $A B C .$

By definition, the arc measure is equal to the measure of the related central angle.

m A B m B C m A B C = m ∠ 1 = m ∠ 2 = m ∠ 3

By the Angle Addition Postulate,

m ∠ 3

can be written as the sum of

m ∠ 1

and

m ∠ 2 .

m A B C = m ∠ 3 ⇕ m A B C = m ∠ 1 + m ∠ 2

Finally, in the above formula,

m A B

and

m B C

can be substituted for

m ∠ 1

and

m ∠ 2,

respectively.

m A B C = m ∠ 1 + m ∠ 2 ⇕ m A B C = m A B + m B C

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

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Proof