Circle Theorem: Alternate Segment Theorem
(KS3, Year 8)

The Lesson

The angle between a tangent to a circle and a chord at the point of contact is equal to the angle in the alternate segment.

This theorem is called the alternate segment theorem.

More About the Alternate Segment Theorem

This circle theorem deals with a tangent and a chord meeting at a point on a circle, forming an angle between them.

The chord divideds the circle into two segments. The segment that does not contain the angle between the tangent and the chord is the alternate segment.

If the ends of the chord are joined to any point on the circle in the alternate segment, the angle between the lines is equal to the angle between the tangent and the chord.

How to Use the Alternate Segment Theorem

Question

What is the angle θ in the circle below?

Step-by-Step:

1

The angle in the alternate segment, θ, is equal to the angle between the tangent and the chord.
θ = 60°

Answer:

The angle in the alternate segment is 60°.

Lesson Slides

The slider below shows a real example of the circle theorem that the angle between a tangent and a chord is equal to the angle in the alternate segment. Open the slider in a new tab

Useful Definitions

A tangent is a line that touches the circle at one point.

A chord is a line whose endpoints lie on the circle.

A segment is a region, not containing the center, bounded by a chord and an arc lying between the chord's endpoints.

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See Also

What is a circle? What are the circle theorems?