| # | Statement | Reason |
|---|---|---|
| 1. | Angle c is 90 degrees | Thales theorem |
| 2. | Angle e is 25 degrees | Inscribed Angle theorem (see Note1) |
| 3. | Angle f is 25 degrees | See "Note2" below |
| 4. | Arc a is 50 degrees | Inscribed Angle theorem 2*(angle f) = (arc a) |
| 5. | Angle d is 65 degrees | d+f = 90, plug in f = 25 and solve for d |
| 6. | Arc b is 80 degrees | See "Note3" below. |
| 7. | Angle k is 90 degrees | Thales theorem |
| 8. | Angle g is 56 degrees | See "Note4" |
| 9. | Angle L is 34 degrees | g+L = 90. Plug in g = 56 and solve for L |
| 10. | Arc n is 112 degrees | Inscribed Angle theorem 2*(angle g) = (arc n) |
Notes:
| Note1 | Cut the 50 degree arc (between L1 and L2) in half to get 25. This is using the inscribed angle theorem. |
| Note2 | Alternate interior angles are congruent (works because L1 || L2), so e = f |
| Note3 | a+b+50 = 180 since the three arcs form a semicircle. Plug in a=50 and solve for b. |
Note4: I couldn't fit this note anywhere else since it's quite long. And it requires a visual aid. Anyways, start by focusing on the angle "112 degrees" given on the original drawing. The vertical angle opposite it is also 112 degrees (see drawing below). The green triangle is an isosceles triangle. I'll save time/space and not provide the proof, but I can add it if you want to see the proof. Because the green triangle is an isosceles triangle, this means that the red angles marked on the same drawing are 34 degrees. I found 34 by solving x+x+112 = 180 for x. The moment I determined that x = 34 degrees, I used it to determine angle g. This is because x+g = 90 degrees (L3 is tangent to the circle, so a right 90 degree angle is formed).
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Summary:
a = 50
b = 80
c = 90
d = 65
e = 25
f = 25
g = 56
h = skipped for some reason
i = skipped for some reason
j = skipped for some reason
k = 90
L = 34
m = skipped for some reason
n = 112
All numbers above in the "summary" are angles in degrees. When I write "skipped for some reason", I mean that the authors of this problem skipped over the letters and I don't have a clue why. For example, they go from 'a' through 'g', skip h,i,j and then use k. I'm not exactly sure of their reasoning here.