Lesson 4 — Inscribed Angles
Big idea: An inscribed angle has its vertex on the circle, and its measure is half the measure of the arc it intercepts.
Words to know
| Word | Meaning |
|---|---|
| Inscribed angle | An angle whose vertex is on a circle and whose sides each intersect the circle in another point. |
| Intercepted arc | The arc inside an angle whose endpoints lie on the angle’s sides. |
| Inscribed in an arc | The angle’s vertex lies on that arc. |
| Semicircle | An arc of 180° whose endpoints are the endpoints of a diameter. |
Concepts to know
- Theorem 63: An inscribed angle is equal in measure to half its intercepted arc.
- Corollary 1: Inscribed angles that intercept the same arc are equal.
- Corollary 2: An angle inscribed in a semicircle is a right angle.
- An inscribed angle and the central angle that intercept the same arc are related: the central angle is twice the inscribed angle.
- Every inscribed angle divides the circle into two arcs: the intercepted arc and the arc the angle is inscribed in.
- Corollary 1 explains why points on the same arc “see” a chord at the same angle.
Formulas / rules
- Inscribed angle = ½ · (measure of intercepted arc)
- Intercepted arc = 2 · (inscribed angle)
- If an inscribed angle intercepts a semicircle, its measure is 90°.
- Central angle = 2 · (inscribed angle intercepting the same arc)
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find an inscribed angle | Halve the measure of its intercepted arc. |
| Find an intercepted arc | Double the inscribed angle. |
| Compare two inscribed angles | If they intercept the same arc, set them equal. |
| Find an angle in a semicircle | It is a right angle, 90°. |
| Find a central angle | Double the inscribed angle that intercepts the same arc. |
| Prove two angles are equal | Show they are inscribed angles intercepting the same arc. |
Watch out
- Halve the arc for an inscribed angle; do not double it.
- The vertex must be on the circle — an angle with its vertex at the center is a central angle.
- Corollary 1 requires the same intercepted arc, not just any arc.
- An angle inscribed in a semicircle is 90° only when the intercepted arc is a semicircle.
- Do not confuse the arc the angle intercepts with the arc it is inscribed in.