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

The line segment AC is tangent to the circle at point B. The points B, D, and E lie on the circumference of the circle, forming the chord BE and the angle $\angle BDE$.

The Alternate Segment Theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
In this figure:

• $\angle CBE$ is the angle between the tangent AC and the chord BE at the point of contact B.
• $\angle BDE$ is the angle subtended by the chord BE in the alternate segment (the segment of the circle opposite to the one containing $\angle CBE$).

According to the Alternate Segment Theorem, the angle between the tangent AC and the chord BE ($\angle CBE$) is equal to the angle subtended by the chord BE in the alternate segment ($\angle BDE$).

Therefore, we have proven that: