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 figure 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 figure 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

Central angles and inscribed angles that subtend the same arc have a special relationship. Let's observe 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

Let's prove the relationship between central angle and inscribed angle by constructing auxiliary lines.

Proof steps:

1. Construct line $CD$ that passes through point $O$ (center of the circle)
2. Note that $OA = OB = OC = OD$ (radii of the circle)
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: $\angle AOB = 2x + 2y = 2(x + y) = 2 \times \angle ACB$

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 $AB$.

   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

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 $\angle ABC = 45^\circ$. Find the measure of central angle $\angle AOB$!

3. In a circle, inscribed 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!

Answer Key

1. Inscribed angle is $\frac{1}{2} \times 120^\circ = 60^\circ$

2. Central angle $\angle AOB = 2 \times 45^\circ = 90^\circ$

3. $\angle PQR = 90^\circ$ (an inscribed angle subtending a diameter is always $90^\circ$)

4. The other angle is $25^\circ$ (inscribed angles subtending the same arc have equal measures)