I wrote this activity to replace a lecture on chords, tangents, and auxiliary lines within circles. I open the lesson with a "mini-lesson" showing them the concept of auxiliary lines (much like problem 1 of the worksheet). After the mini-lesson, I hand out the worksheet and set them to work, which allows 45 minutes to an hour during which the students can work cooperatively together as they work on these challenging problems.
Before you use this worksheet, your students should be familiar with inscribed angles vs. central angles, intercepted arcs, and basic vocabulary for circles: major arcs, minor arcs, chords, and so forth.
I include review of the Exterior Angle Theorem (EAT), as that is a concept taught in the first semester at my school, and by second semester, I don't necessarily have the same batch of kids. I use an analogy of a "monster" located at one exterior angle of a triangle, and I ask them where they'd run if a monster were pursuing them, and we end up deciding, basically, that you'd locate yourself at the two remote interior angles of a triangle. (Of course, they never use that terminology, but I incorporate it into the discussion).
The file includes spiral review problems.
This worksheet is intended to be written on directly.
Please download the pdf preview file first, so you can see exactly what's included; the product file is a word document, which you may personalize for your students.