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source: http://www.onlinemathlearning.com

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

*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*.