Alternate Angle Segment Theorem Proof

The theorem states that the angle between the tangent and its chord is equal to the angle in the alternate segment.

Alternate Angle Segment Theorem

Statement:

An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment

Diagram:

Proof:

A tangent makes an angle of 90 degree with the radius of the circle.So, we know that <OAC + x = 90.
The angle in a semi-circle is 90, so <BCA = 90.

The angles in a triangle add up to 180, so <BCA + <OAC + y = 180

Therefore 90 + <OAC + y = 180 and so <OAC + y = 90
But OAC + x = 90, so <OAC + x = <OAC + y

Hence x = y
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