Question

explore: tangent and chord
what do you notice? where is the vertex for each angle?

Explanation:

For the angle formed by a tangent and a chord (tangent - chord angle), the vertex of each angle lies on the circle (specifically, the point of tangency where the tangent touches the circle). Observing the diagrams, we can see that in each case, the angle is formed at the point where the tangent line meets the circle, and the chord connects that point to another point on the circle. Also, there is a relationship between the measure of the tangent - chord angle and the measure of the intercepted arc. The measure of the tangent - chord angle is half the measure of its intercepted arc. For example, if we look at the arc opposite to the angle (the arc that is "cut off" by the chord and the tangent), the angle's measure is half of that arc's measure.

Answer:

The vertex of each angle (formed by a tangent and a chord) is at the point of tangency (on the circle). Also, the measure of the tangent - chord angle is half the measure of its intercepted arc.