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Circle Theorem: The Alternate Segment Theorem (Mathematics Lesson)

Circle Theorem: The Angle Between a Tangent and a Chord is Equal to the Angle in the Alternate Segment

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.



It is sometimes 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?



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

θ = 60°

A Real Example of How to Use the Alternate Segment Theorem

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:
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Note
WHAT IS A CIRCLE?

A circle is a shape containing a set of points that are all the same distance from a given point, its center.

CIRCLE THEOREMS

Circle theorems relate to the angles and lines within circles.

That the angle between a tangent and a chord is equal to the angle in the alternate segment is one of the circle theorems. There are several others.

Read more about circle theorems

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.