Circle Theorems

In this lesson, we will learn

• inscribed angles.
• a circle theorem about inscribed angles which is sometimes called the Bow Theorem. It states that the inscribed angles subtended by the same arc or chord are equal.

Inscribed Angles

We will first look at what is meant by inscribed angle or angle at the circumference.

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

∠ABC is also called an angle at the circumference.

Bow Theorem

The theorem states

The inscribed angles subtended by the same arc or chord are equal.

or

Arcs that contain equal angles are equal.

or

Angles in the same segment are equal.

or

The angles at the circumference subtended by the same arc are equal.

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

You can see from the above diagram that the inscribed angles form a "bow-tie". That is why it is sometimes called the bow theorem.

Example:

Find the value of ∠ x in the figure below.

Solution:

∠ x = 38˚ because they are both subtended by the same arc PRQ.

This video explains the inscribed angle property and how it can be used to find missing angles.
(1) Inscribed angles from equal arcs are equal.
(2) Arcs that contain equal angles are equal.

This video shows the two useful properties when solving problems that involve inscribed angles.
(1) The measure of the inscribed angle is half the measure of the central angle.
(2) Inscribed angles that intercept congruent arcs are congruent.

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