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

Related Topics: More Circle Theorems and Geometry Lessons

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.

inscribed angles

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

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

angle semicircle

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.





Central angles and inscribed angles are two types of angles that relate to circles.
A central angle has its vertex is the middle of the circle.
An inscribed angle has one endpoint on the edge of the circle and then cuts across the rest of the circle. The vertex of its angle is on the circumference.

Any inscribed angle that ends on the same two points has the same measure unless the vertex is on the minor arc. If their vertices are on opposite arcs then they are supplementary. 
Any inscribed angle whose endpoints are a diameter is a right angle, or 90 degree angle.

The Central Angle Theorem states that the inscribed angle is half the measure of the central angle. In this video, we can see that the purple inscribed angle and the black central angle share the same endpoints. If the inscribed angle measure x, the central angle will measure 2x. For example, if the central angle is 90 degrees, the inscribed angle is 45 degrees.






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.

inscribed angle


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.






How to solve problems involving angles inscribed in circles.



Examples of using the Central Angle Theorem to find missing angles




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.



Proof of Central Angle Theorem (which would also the proof the Inscribed Angles Theorem)
The central angle is twice the inscribed angle that subtends the same arc.






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