What is a secant angle?

Outside the Circle Crew: Imagine two secant lines meeting outside the circle, like friends catching up at a cafe. The angle they create is special: it’s half the difference between the arcs they “cut off” on the circle. Picture a pizza. The arcs are the crust. If one arc is, say, 80 degrees and the other is 30 degrees, you subtract them (80 – 30 = 50) and then halve that (50 / 2 = 25). Boom! The angle is 25 degrees.

Secant and Tangent Tango: Now, picture a secant and a tangent meeting outside the circle. It’s the same deal as before. You still take half the difference of the intercepted arcs to find the angle.

Inside Job (Chord Chaos): Okay, things change a bit when the lines (now technically chords, since they stop at the circle’s edge) intersect inside the circle. Instead of subtracting, we add the intercepted arcs and then halve the result. So, if the arcs are 60 and 40 degrees, you add them (60 + 40 = 100) and halve it (100 / 2 = 50). The angle is 50 degrees.

Tangent-Chord Huddle: When a tangent and a chord meet on the circle, the angle they create is simply half the measure of the arc they trap. Easy peasy!

Secant-Secant Power Play: Imagine two secants drawn from the same spot outside the circle. The Secant-Secant Power Theorem says that if you multiply the length of one entire secant by the length of its outside part, it’s the same as doing that for the other secant. It’s like a cosmic balance!

Secant-Tangent Showdown: If you have a tangent and a secant coming from the same point outside the circle, the square of the tangent’s length equals the length of the whole secant multiplied by its external part. Pretty neat, huh?

Intersecting Secants Theorem: When two secants intersect at an exterior point, the product of the one whole secant segment and its external segment is equal to the product of the other whole secant segment and its external segment. This is also known as the secant theorem or the secant power theorem.