Hi geometric,

There is quite a lot of explanation that should accompany these answers. I hope you've studied your circle theorems.

(a) because you were given that , which is half of the intercepted arc BC.

(b) because you can find the measure of arc AE by subtracting the other known arcs from 360. Arc AE measures 80 degrees. That makes the inscribed angle = 40.

(c) because the angle formed by the intersection of two chords is one-half the sum of the measures of the two subtended arcs. Arc AE = 80 degrees and arc BC = 100 degrees. Their sum is 180, and half that is 90 degrees.

(d) because the exterior angle formed by two secants is equal to one-half the difference of its intercepted arcs. Thus angle F is 1/2(150 - 80) or 35 degrees.

(e) since it is in the right triangle FAD and is complementary to angle F.

Your second question requires even more explanation. I suggest you post it separately and tell us how far along you have gotten. Be specific about what you don't understand.

Study your circle theorems (all about central angles, inscribed angles, tangents, secants, chords, arcs, etc.).