intercepted arc — Definition, Formula & Examples
An intercepted arc is the portion of a circle that lies between the two sides of an inscribed angle, central angle, or any angle whose sides intersect the circle. The endpoints of the arc are the points where the angle's sides meet the circle.
Given an angle whose vertex lies on, inside, or outside a circle and whose sides (or their extensions) intersect the circle at two distinct points, the intercepted arc is the arc of the circle whose endpoints are those two intersection points and which lies in the interior of the angle.
How It Works
To identify an intercepted arc, start at the vertex of the angle and follow each side until it hits the circle. The two points where the sides meet the circle are the arc's endpoints. The arc that sits inside the angle is the intercepted arc. For a central angle, the intercepted arc has the same degree measure as the angle. For an inscribed angle, the intercepted arc has twice the degree measure of the angle.
Worked Example
Problem: An inscribed angle in a circle measures 35°. What is the degree measure of its intercepted arc?
Recall the Inscribed Angle Theorem: An inscribed angle equals half the measure of its intercepted arc.
Solve for the arc: Multiply both sides by 2 and substitute the given angle measure.
Answer: The intercepted arc measures 70°.
Why It Matters
Intercepted arcs connect angle measures to arc measures, which is essential for solving problems in high-school geometry involving inscribed angles, tangent-chord angles, and secant-secant angles. Understanding them is also necessary for proofs involving cyclic quadrilaterals and circle theorems on the SAT and ACT.
Common Mistakes
Mistake: Confusing the intercepted arc with the entire remaining arc of the circle.
Correction: The intercepted arc is only the arc that lies in the interior of the angle, not the arc on the exterior side. For an inscribed angle of 35°, the intercepted arc is 70°, not 290°.