The Inscribed Angle Theorem
📐 Key Definitions
- Central Angle
- An angle whose vertex is at the center of the circle and whose sides are radii. The central angle equals the measure of its intercepted arc.
- Inscribed Angle
- An angle whose vertex is on the circle and whose sides are chords of the circle.
- Intercepted Arc
- The arc that lies in the interior of the angle and has its endpoints on the angle's sides.
🌟 The Inscribed Angle Theorem
$$ \text{Inscribed Angle} = \frac{1}{2} \times \text{Intercepted Arc} $$
An inscribed angle is exactly half the measure of its intercepted arc. Equivalently, an inscribed angle is half the central angle that subtends the same arc.
Why Does This Work? (Intuition)
Imagine a central angle $\angle AOB$ where $O$ is the center. This angle equals the arc $\overset{\frown}{AB}$. Now place a point $C$ on the circle. The inscribed angle $\angle ACB$ "sees" the same arc but from farther away — from the circumference instead of the center. Because the vertex is on the circle rather than at the center, the angle is exactly half as wide.
This can be proven using the isosceles triangle formed by two radii and a chord. Since $OA = OC$ (both radii), triangle $OAC$ is isosceles, and the base angles are equal. Using the exterior angle theorem on the resulting triangles gives us the factor of $\frac{1}{2}$.
Important Corollaries
Corollary 1
Inscribed angles that intercept the same arc are equal.
If $\angle ACB$ and $\angle ADB$ intercept $\overset{\frown}{AB}$, then $\angle ACB = \angle ADB$.