Circles have many interesting geometric properties. In these lessons, 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.

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

The following diagram shows another example of the alternate segment theorem.

* 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.

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

The angle between a chord and a tangent is equal to the angle drawn from the same chord (this is in the alternate segment of the circle).

These video shows how to use the alternate segment theorem to find missing angles.

This exam question involves using knowledge of the alternate segment theorem, tangent and radius theorem and knowledge of angles in an isosceles triangle to find a missing angle in a geometric scenario involving circles and tangents.

How to solve an exam question that uses the Alternate Segment Theorem and properties of a Tangent?

(Maths GCSE Revision)

This video involves looking at a couple of circle theorem questions. We use knowledge of the tangent meeting the radius at 90 degrees, the angle in a semi-circle being 90 degrees, the alternate segment theorem, opposite angles in a cyclic quadrilateral being equal to 180 degrees and other basic angle facts involving triangles.

Draw 3 radii from the center of the circle to the 3 points on the circle to form 3 isosceles triangles.

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

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