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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.

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

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