How do you apply the tangent-chord angle theorem?

To apply the tangent-chord angle theorem, identify the tangent and chord, then use the theorem to find the angle.

The tangent-chord angle theorem states that the angle formed between a tangent to a circle and a chord through the point of contact is equal to the angle in the alternate segment. To use this theorem, first identify the tangent line and the chord that meet at a point on the circle. The point where the tangent touches the circle is called the point of tangency.

Next, locate the angle formed between the tangent and the chord. This is the angle you will be working with. According to the theorem, this angle is equal to the angle in the alternate segment of the circle. The alternate segment is the part of the circle that lies opposite the chord, and the angle in this segment is formed by drawing two lines from the ends of the chord to another point on the circle.

For example, if you have a circle with a tangent at point A and a chord AB, the angle between the tangent and the chord (let's call it ∠TAB) is equal to the angle in the alternate segment, which is the angle subtended by the chord AB at the opposite side of the circle (let's call it ∠ACB). By identifying these angles, you can use the theorem to solve problems involving circle geometry, such as finding unknown angles or proving that two angles are equal.