Central and Inscribed Angles
Use the relationship between the two kinds of angles in a circle:
- A central angle has its vertex at the center of the circle.
- An inscribed angle has its vertex on the circle.
The key idea is that an inscribed angle that intercepts the same arc as a central angle has half the measure of the central angle.
Method
- Identify the angle type: decide whether the given angle is central or inscribed.
- Find the intercepted arc or matching angle: look for the arc both angles intercept.
- Apply the circle relationship:
- inscribed angle = half of the corresponding central angle
- central angle = twice the corresponding inscribed angle
- Solve for the unknown and simplify the final answer.
Useful checks
- Central angles should match the measure of their intercepted arcs.
- Inscribed angles should be smaller than the matching central angle, because they are half as large.
- If several angles intercept the same arc, they should all give consistent results.
Tip
When an expression is involved, set up the equation first, then isolate the variable carefully. After solving, substitute back to confirm the angle relationship is correct.