Answer:
1 = 60, 4 = 30, 5 = 30, 6 = 150, 7 = 110, 8 = 30, 9 = 60, 10 = 90. 2 = 47.5, 3 = 132.5
Explanation:
Starting with angle 5: knowing that arc BC = 30 and angle 5 is a central angle, it has an angle measure equal to the measure of the arc it intercepts.
Angle 6 is supplementary to angle 5, so 180 - 30 = 150.
Angle 9 = 60. Ch is a diameter, so that means it splits the circle into 2 congruent halves, each measuring 180 degrees around the outside. So if arcs AB and CB both measure 30, then arc AH measures 180 - 30 - 30 = 120. By definition, the measure of angle 9 is half the measure of the arc it intercepts.
Angles 4 and 8 = 30 each. Because you have AH parallel to CG, then CH is a transversal, creating a pair of alternate interior angles that are congruent. Those angles are 4 and 8. We find 8 to be an inscribed angle, cutting off arc ABC which measures 60 degrees. An inscribed angle is half the measure of the arc it intercepts. Because angle 4 measures 30 degrees and is inscribed, the arc it cuts off, arc GH, measures twice the angle cutting it. So arc GH measures 60.
Again, since CH is a diameter, then the semicircle CDEG measures 180. Since we know from the description that arcs CD and DE are congruent, then arc CD + arc DE + arc EG (given as 50) + arc GH = 180. Since arcs CD and DE are congruent, lets just call them "x" and we have two of them. That gives us that 2x + 50 + 60 = 180. x = 35. Angle 1 is equal to half of the sum of its intercepted arcs ( 35 + 35 + 50) which is 60.
Angle COE intercepts arc CDE, and is central, so angle COE measures 70, and since angle 7 is supplemetray to angle COE, then angle 7 measures 180 - 70 = 110.
Angle 10 by definition is a right angle (refer to the theorem regarding a tangent line to a point on a circle).
Angle 2 is half of the sum of 35 (arc CD) and 60 (arc GH), so angle 2 measures 47.5 and that means that angle 3, supplementary to angle 2, measures 180 - 47.5 = 132.5. I think those are all correct. The only one I'm unsure of is angle 2.