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There's an error with Angle CLG = 20°
Pay careful attention to the fact there isn't a segment connecting points C and L.
Angle CLG should be renamed to angle ALG.
Other names for this angle are possible (such as BLG or DLG).
Triangle CIH is a right triangle. More specifically, it is a 30-60-90 triangle.
Triangles CIH and JIH are congruent due to the hypotenuse leg (HL) theorem, which means JIH is also a 30-60-90 triangle.
So angle CJL = 30°
Draw a segment from J to L.
Minor arc CL = 60°, half of that is 30°, which is the measure of angle CJL.
Refer to the inscribed angle theorem.
Angle CJK = 45° (given) and angle CJL = 30°.
This must mean angle LJN = 15° (because 45-30 = 15)
Note that angles LJN and CJL glue together to form angle CJK.
Now focus your attention on triangle LJN only.
We found angle J = 15° just now.
Angle L = 90° since point L is a point of tangency.
Therefore angle N is:
L+J+N = 180
N = 180-L-J
N = 180-90-15
N = 75°
A slight renaming shows that angle LNJ = 75°
This must mean angle JNQ = 180°-75° = 105°
because angles LNJ and JNQ are supplementary.
In the previous problem we found that angle CJL = 30°.
We also determined that triangle JIH is a 30-60-90 triangle with angle JIH = 60°.
Central angle JIH = 60° leads to minor arc GJ = 60°.
The arc subtended by the central angle are congruent measures.
Apply the inscribed angle theorem to determine that angle GLJ = 60/2 = 30°
This is angle L of triangle ELI.
Because arc CL = 60°, it means central angle CIL = 60° as well.
This represents angle I of triangle ELI.
Focus on Triangle ELI
angle E = ??
angle L = 30
angle I = 60
E+L+I = 180
E+30+60 = 180
E+90 = 180
E = 180-90
E = 90
This means angle IEL = 90, and so is angle CEG (as they are vertical angles).
Any pair of vertical angles are always congruent.
angle GIL = angleCIG + angleCIL
angle GIL = 60° + 60°
angle GIL = 120°
This represents angle I of triangle LGI.
Focus on triangle LGI
angle L = 30° (found in problem 2)
angle G = ??
angle I = 120°
L+G+I = 180°
30 + G + 120° = 180°
G+150° = 180°
G = 180° - 150°
G = 30°
This represents angle LGI.
A thing to notice: triangle LGI is isosceles because the radii GI = IL are congruent.
Therefore, the base angles G and L are congruent.
The base angles are opposite the congruent sides.
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