Mess around with the applet below for a few minutes.
Feel free to move the BIG POINTS around before you start dragging the slider.
Be sure to drag the slider slowly and pay careful attention to what happens.
After you drag the slider to the end, press "Reset Slider".
Change the location of the BIG RED POINT and the BIG PURPLE POINT.
Then move the slider slowly again.
Have fun with this applet, and then answer the questions that appear below the applet.

The purple angle (you see when you drag the slider to the end) is called an angle formed by a secant and a tangent.
Given what you've just observed, answer the following questions:
Suppose the red arc measures 150 degrees and the green arc measures 80 degrees.
If this were so, what would the measure of the purple angle be?
Suppose the red arc measures 120 degrees and the green arc measures 30 degrees.
If this were so, what would the measure of the purple angle be?
If the measure of the purple angle was 30 degrees and the measure of the red arc was 190 degrees, what would the measure of the green arc be?
Now, drag the red point to the right. Go as far as you can. What happens?
Why did the "secant" ray disappear?
From your observations, describe how to find the measure of an angle formed by a secant and a tangent. Be sure to use the phrase "intercepted arc(s)" at least once in your description. Please be as specific as you can.