Circle Theorems

In this lesson, we will learn

• inscribed angles and central angles.
• a circle theorem called The Inscribed Angle Theorem or The Central Angle Theorem or The Arrow Theorem.
• how to prove the Inscribed Angle Theorem

Inscribed Angles and Central Angles

We will first look at some definitions.

An inscribed angle has its vertex on the circle. ∠ABC, in the diagram below, is called an inscribed angle or angle at the circumference. The angle is also said to be subtended by (i.e. opposite to) arc ADC or chord AC

Property: The inscribed angles subtended by the same arc are equal.

∠ x = ∠ y because they are subtended by the same arc AED.

 

Property: The inscribed angles in a semicircle is 90˚.

POQ is the diameter. ∠PAQ = ∠PBQ = ∠PCQ = 90˚.

A central angle has its vertex is at the centre of the circle. In the diagram below, ∠AOC is called a central angle.

Property: Central angles subtended by arcs of the same length are equal.

 

∠ x = ∠ y because arc AB = arc CD

This video introduces inscribed angles and central angles.
(1) Central angles subtended by arcs or chords of the same length are equal.
(2) If two inscribed angles subtend the same arc or chord, then the angle measures are equal.

Inscribed Angle Theorem

Now, we will look at the Inscribed Angle Theorem. It is also called the Central Angle Theorem or Arrow Theorem.

The theorem states

The measure of the central angle is equal to twice the measure of the inscribed angle subtended by the same arc

or

An inscribed angle is half of a central angle that subtends the same arc.

or

The angle at the centre of a circle is twice any angle at the circumference subtended by the same arc.

The following diagrams illustrates the Inscibed Angle Theorem

Example:

The center of the following circle is O. BOD is a diameter of the circle. Find the value of x.

Solution:

∠BOC + 70˚ = 180˚

∠BOC= 110˚

2x = 110˚

x = × 110˚

= 55˚

Circle Theorem
Basic definitions: Chord, segment, sector, tangent, cyclic quadrilateral.
Theorem: Angle subtended at the centre of a circle is twice the angle at the circumference.

This video will explain the Central Angle Theorem and how it can be used to find missing angles. It also shows the Central Angle Theorem Corollary: The angle inscribed in a semicircle is a right angle.

Proof of the Inscribed Angle Theorem

This video proves the theorem that an inscribed angle is half of a central angle that subtends the same arc.

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