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adjacent arcs — Definition, Formula & Examples

Adjacent arcs are two arcs on the same circle that share exactly one common endpoint and do not overlap. Together, they form a single larger arc.

Two arcs of a circle are adjacent if and only if they lie on the same circle, intersect at exactly one point (a shared endpoint), and have no interior points in common. The measure of the arc formed by two adjacent arcs equals the sum of their individual measures.

Key Formula

mAC^=mAB^+mBC^m\widehat{AC} = m\widehat{AB} + m\widehat{BC}
Where:
  • mAB^m\widehat{AB} = Degree measure of the first arc
  • mBC^m\widehat{BC} = Degree measure of the second (adjacent) arc
  • mAC^m\widehat{AC} = Degree measure of the combined arc

Worked Example

Problem: On a circle, point B lies between points A and C along the circumference. Arc AB measures 70° and arc BC measures 110°. Find the measure of arc AC.
Identify the adjacent arcs: Arcs AB and BC share exactly one endpoint (B) and do not overlap, so they are adjacent.
Apply the Arc Addition Postulate: Add the measures of the two adjacent arcs to find the combined arc.
mAC^=mAB^+mBC^=70°+110°=180°m\widehat{AC} = m\widehat{AB} + m\widehat{BC} = 70° + 110° = 180°
Answer: Arc AC measures 180°, making it a semicircle.

Why It Matters

The Arc Addition Postulate — which depends on arcs being adjacent — is used throughout high school geometry whenever you calculate arc lengths, sector areas, or inscribed angle measures. It works much like the Angle Addition Postulate but applied to curved portions of a circle.

Common Mistakes

Mistake: Assuming two arcs on the same circle are adjacent even when they overlap.
Correction: Adjacent arcs must share exactly one endpoint and have no interior points in common. If arcs overlap (share more than one point), they are not adjacent and you cannot simply add their measures.

Related Terms