Central & Inscribed Angles
- Relate central angles, inscribed angles, and intercepted arcs.
- Find arc measures using angle relationships.
- Apply central and inscribed angle rules in circle diagrams.
Key Ideas
Central Angle
A central angle has its vertex at the center of the circle.
The measure of a central angle equals the measure of its intercepted arc.
\[ m(\angle_{\text{central}}) = m(\text{intercepted arc}) \]
Inscribed Angle
An inscribed angle has its vertex on the circle.
The measure of an inscribed angle equals half the measure of its intercepted arc.
\[ m(\angle_{\text{inscribed}}) = \frac12 m(\text{intercepted arc}) \]
Central vs. Inscribed Angles
If a central angle and an inscribed angle intercept the same arc, then the central angle is twice the inscribed angle.
\[ m(\angle_{\text{central}}) = 2m(\angle_{\text{inscribed}}) \]
Equivalently,
\[ m(\angle_{\text{inscribed}}) = \frac12 m(\angle_{\text{central}}) \]
Common Problem Types
Central Angle from Arc
Use direct equality.
If the intercepted arc is \(120^\circ\), then the central angle is also \(120^\circ\).
Inscribed Angle from Arc
Use the half-rule.
If the intercepted arc is \(80^\circ\), then the inscribed angle is:
\[ \frac12(80^\circ)=40^\circ \]
Arc from Inscribed Angle
Double the inscribed angle.
If the inscribed angle is \(35^\circ\), then the intercepted arc is:
\[ 2(35^\circ)=70^\circ \]
Inscribed Angle from Central Angle
If both angles intercept the same arc, the inscribed angle is half the central angle.
If the central angle is \(100^\circ\), then the inscribed angle is:
\[ \frac12(100^\circ)=50^\circ \]
Special Cases
- An angle inscribed in a semicircle is \(90^\circ\).
- Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
Strategies
- Identify whether the angle is central or inscribed.
- Trace the intercepted arc carefully.
- If two angles intercept the same arc, compare them directly:
- central angle = twice the inscribed angle
- inscribed angle = half the central angle
- For semicircles, use \(180^\circ\) arc \(\rightarrow\) inscribed angle \(=90^\circ\).
Worked Examples
Example 1
An inscribed angle intercepts an arc measuring \(80^\circ\). Find the measure of the inscribed angle.
Since an inscribed angle is half the measure of its intercepted arc:
\[ \frac12(80^\circ)=40^\circ \]
So the inscribed angle measures:
\[ 40^\circ \]
Example 2
An inscribed angle measures \(35^\circ\). Find the measure of its intercepted arc.
Since the intercepted arc is twice the inscribed angle:
\[ 2(35^\circ)=70^\circ \]
So the intercepted arc measures:
\[ 70^\circ \]
Example 3
A central angle and an inscribed angle intercept the same arc. The central angle measures \(110^\circ\). Find the measure of the inscribed angle.
Since the inscribed angle is half the central angle:
\[ \frac12(110^\circ)=55^\circ \]
So the inscribed angle measures:
\[ 55^\circ \]
- Using the half-rule on central angles.
- Saying an inscribed angle is half the arc length instead of half the arc measure.
- Mixing up which arc an angle intercepts.
- Forgetting that a central angle is twice an inscribed angle when both intercept the same arc.
- Forgetting the semicircle rule when a diameter is present.
Practice Problems
- A central angle measures \(110^\circ\). What is the measure of its intercepted arc?
- An inscribed angle measures \(25^\circ\). What is the measure of its intercepted arc?
- An angle is inscribed in a semicircle. What is the measure of the angle?
- A central angle and an inscribed angle intercept the same arc. If the central angle is \(96^\circ\), what is the inscribed angle?
- A central angle and an inscribed angle intercept the same arc. If the inscribed angle is \(42^\circ\), what is the central angle?
- A central angle equals the measure of its intercepted arc.
\[ 110^\circ \]
- The intercepted arc is twice the inscribed angle.
\[ 2(25^\circ)=50^\circ \]
- An angle inscribed in a semicircle is a right angle.
\[ 90^\circ \]
- The inscribed angle is half the central angle.
\[ \frac12(96^\circ)=48^\circ \]
- The central angle is twice the inscribed angle.
\[ 2(42^\circ)=84^\circ \]
Summary
- Central angle = intercepted arc measure.
- Inscribed angle = half the intercepted arc measure.
- If both intercept the same arc, central angle = twice the inscribed angle.
- Semicircle \(\rightarrow\) inscribed angle \(=90^\circ\).
- If the vertex is at the center, use direct equality.
- If the vertex is on the circle, use the half-rule.
- If a central and inscribed angle intercept the same arc, compare them directly.