Circle Theorems

In this lesson, we will learn

• a Circle Theorem called The Alternate Segment Theorem.
• how to use the alternate segment theorem.
• how to prove the alternate segment theorem.

Alternate Segment Theorem

The Alternate Segment theorem states

An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Recall that a chord is any straight line drawn across a circle, beginning and ending on the curve of the circle.

In the following diagram, the chord CE divides the circle into 2 segments. Angle CEA and angle CDE are angles in alternate segments because they are in opposite segments.

The alternate segment theorem states that an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment

In terms of the above diagram, the alternate segment theorem tells us that angle CEA and angle CDE are equal.

Example:

In the following diagram, MN is a tangent to the circle at the point of contact A. Identify the angle that is equal to x

Solution:

We need to find the angle that is in alternate segment to x.

x is the angle between the tangent MN and the chord AB.

We look at the chord AB and find that it subtends angle ACB in the opposite segment.

So, angle ACB is equal to x.

This video shows how to identify the angles that are equal for the alternate segment theorem: the angle between a tangent and a chord is equal to the angle in the alternate segment.

Use of the Alternate Segment Theorem

This videos show how to use the alternate segment theorem to find missing angles.

Proof for the Alternate Segment Theorem

This video will show how to prove the alternate segment theorem.

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