Alternate Angle Segment Theorem Proof

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