Final answer:
An angle formed by a tangent and chord or secant is always half the measure of the intercepted arc, which applies to geometry problems involving circles.
Step-by-step explanation:
When an angle is formed by a tangent and a chord or secant, the angle is always half the measure of the intercepted arc. This concept is a fundamental aspect of circle theorems. To illustrate, when a tangent and chord intersect at a point on a circle, the angle formed between them at that point is exactly half the measure of the arc that lies between the points of contact. This is because the tangential line forms an angle of 90 degrees with the radius, creating two right-angle triangles which share the intercepted arc as their hypotenuse, and hence the connection with the arc's measurement.
Considering a tangent and a secant, which is an extended chord, they cut the circle at two points. Similar to the chord, the angle made at the point of tangency is half the measure of the arc between those two points on the circumference. This relationship between tangents, chords, secants, and their corresponding arcs can help solve various geometrical problems involving circles.