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Circle Theorems - Alternate Segment Theorem




 

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.

Related Topics: More Geometry Lessons

What is 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.

alternate segment theorem

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.
alternate segment theorem




How to use the Alternate Segment Theorem to find missing angles?

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.



 
This video explains what the alternate segment theorem is. 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).
Use of the Alternate Segment Theorem
These videos show how to use the alternate segment theorem to find missing angles.
This 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.


Circle Theorem Exam Question - Alternate Segment and Tangent.
Circle theorems and the Alternate Segment Theorem - (Maths GCSE Revision)


 
Circle Theorems (aiming for a B grade and beyond)
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.
How to proof the Alternate Segment Theorem
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.
Proof of the Alternate Segment Theorem in Circle Theorems


Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.


You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.


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