Circle Theorem: Alternate Segment Theorem
(KS3, Year 8)
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. circle theorem angle in alternate segment is equal 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. tangent_chord_angle 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. 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. alternate_segment_angles

How to Use the Alternate Segment Theorem

Question

What is the angle θ in the circle below?
alternate segment example

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.

Useful Definitions

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

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

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

circle segment
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This page was written by Stephen Clarke.