The angle formed between a tangent and a chord is equal to the angle in the alternate segment of the circle.

When a tangent touches a circle at a specific point, and a chord is drawn from that point, the angle formed between the tangent and the chord is referred to as the angle between the tangent and the chord. This particular angle is equal to the angle in the alternate segment of the circle. The alternate segment is defined as the area of the circle that lies opposite to the chord.

To illustrate this concept, consider a circle with a tangent line touching it at point $A$. Next, draw a chord $AB$ from point $A$ to another point $B$ on the circumference of the circle. The angle between the tangent at point $A$ and the chord $AB$ is equal to the angle subtended by the chord $AB$ at the circumference of the circle on the side opposite to the chord. This angle is located in the alternate segment.

This property is derived from the Alternate Segment Theorem, which asserts that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. This theorem proves to be exceptionally useful for solving a variety of problems related to circles, particularly when determining unknown angles.

For instance, if you are given the angle between the tangent and the chord, you can easily deduce the angle in the alternate segment, and vice versa. This relationship streamlines the process of analyzing and solving geometric problems that involve circles, tangents, and chords, making it a valuable tool in geometry.