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:

• Measure of inscribed angle = $\frac{1}{2}$ × measure of central angle
• Measure of central angle = 2 × 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° (right angle).

   Inscribed Angle Subtending a Diameter

   Inscribed angle subtending a diameter is always 90°.

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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 AOB = 80°. Find the measure of inscribed angle ACB that subtends the same arc!

Solution:

Calculating Central Angle

Given inscribed angle ACB = 35°. Find the measure of central angle AOB that subtends the same arc!

Solution:

Practice Problems

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

2. Inscribed angle ABC = 45°. Find the measure of central 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°, find the measure of the other angle!

Answer Key

1. Inscribed angle = $\frac{1}{2} \times 120° = 60°$

2. Central angle AOB = $2 \times 45° = 90°$

3. Angle PQR = 90° (inscribed angle subtending a diameter is always 90°)

4. The other angle = 25° (inscribed angles subtending the same arc have equal measures)