Central Angle and Inscribed Angle

Definition of Central Angle

A central angle is an angle formed by two radii of a circle with the vertex located at the center of the circle. The sides of the central angle are radii that connect the center to points on the circle.

In the diagram above:

• Point $O$ is the center of the circle
• $OA$ and $OB$ are radii of the circle
• $\angle AOB$ is the central angle
• The measure of the central angle is denoted by $\alpha$

Definition of Inscribed Angle

An inscribed angle is an angle formed by two chords with the vertex located on the circle. The sides of the inscribed angle are chords that connect the vertex to two other points on the circle.

In the diagram above:

• Point $C$ is located on the circle
• $CA$ and $CB$ are chords
• $\angle ACB$ is the inscribed angle
• Point $O$ is the center of the circle

Relationship Between Central Angle and Inscribed Angle

When they subtend the same arc, the central angle is twice the inscribed angle. The diagram below shows this relationship.

Theorem of Central Angle and Inscribed Angle Relationship

If a central angle and an inscribed angle subtend the same arc, then:

• $\text{measure of inscribed angle} = \frac{1}{2} \times \text{measure of central angle}$
• $\text{measure of central angle} = 2 \times \text{measure of inscribed angle}$

Proof of Central Angle and Inscribed Angle Relationship

Draw the diameter through $C$ and the center $O$. The two resulting isosceles triangles connect the inscribed angle to the central angle.

Proof steps:

1. Construct line $CD$ that passes through point $O$ (center of the circle)
2. Because all four segments are radii, $OA = OB = OC = OD$
3. Triangles $AOC$ and $BOC$ are isosceles triangles
4. Let $\angle ACO = x$ and $\angle BCO = y$
5. Since they are isosceles triangles: $\angle CAO = x$ and $\angle CBO = y$
6. Exterior angles of triangles: $\angle AOD = 2x$ and $\angle BOD = 2y$
7. Therefore: $\angle AOB = 2x + 2y = 2(x + y) = 2 \times \angle ACB$

This construction proves the configuration shown, where $C$ lies on the arc opposite $AB$. Other positions can be split into the same isosceles-triangle cases. In every case, the two angles being compared must intercept the same arc.

Properties of Central Angle and Inscribed Angle

1. Inscribed Angle Subtending a Diameter

   Every inscribed angle that subtends a diameter of a circle measures $90^\circ$ (right angle).

   Inscribed Angle Subtending a Diameter

   Inscribed angle subtending a diameter is always $90^\circ$.

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2. Inscribed Angles Subtending the Same Arc

   $\angle ACB = \angle ADB$, both angles subtend the same arc $AB$. In other words, sliding $C$ to $D$ along the same arc does not change the measure.

   Inscribed Angles Subtending the Same Arc

   Inscribed angles that subtend the same arc have equal measures.

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Calculating Inscribed Angle

Given central angle $\angle AOB = 80^\circ$. Find the measure of inscribed angle $\angle ACB$ that subtends the same arc!

Solution:

Calculating Central Angle

Given inscribed angle $\angle ACB = 35^\circ$. Find the measure of central angle $\angle AOB$ that subtends the same arc!

Solution:

Practice Problems

Each problem gives either the central angle or the inscribed angle and asks for the other. Both angles subtend the same arc.

1. If the central angle of a circle is $120^\circ$, what is the measure of the inscribed angle that subtends the same arc?

2. Inscribed angle $\angle ACB = 45^\circ$. Find the measure of central angle $\angle AOB$ that intercepts the same arc.

3. In a circle, inscribed angle $\angle PQR$ subtends a diameter. What is the measure of $\angle PQR$?

4. Two inscribed angles subtend the same arc. If one angle measures $25^\circ$, find the measure of the other angle!

Worked Solutions

1. The inscribed angle subtends the same arc as the central angle, so its measure is half the central angle: $\frac{1}{2} \times 120^\circ = 60^\circ$.

2. The central angle and $\angle ACB$ intercept arc $AB$. The central angle is twice the inscribed angle:

   $$\angle AOB=2\angle ACB=2(45^\circ)=90^\circ$$

3. A diameter cuts off a semicircle, whose central angle is $180^\circ$. The inscribed angle is half of that angle, so $\angle PQR = 90^\circ$.

4. Inscribed angles that intercept the same arc have equal measures. The other angle is also $25^\circ$.