Now to finish this off:
https://image.ibb.co/hWvF2f/Stage-4.png
I had to restore the measures of some of the original angles (angles CAE and EAF) to clarify things here, and hopefully it's clear what angles correspond to which regions as indicated by the matches in colors. If still confusing, let me know. This solution is a bit of a mess, but I've tried to make this as clean as possible.
In this stage, one more line had to be drawn, a line from C to G which just happens to bisect angle ACB (remember that its measure is 20). If not convinced, work out how to prove triangles CAG and CBG are SSS-congruent by remembering that ABG is an equilateral triangle. Should be easy.
Consequently, all new angles in the image above should be easy to figure out on your own, so I won't spend too much energy trying to show how their measures have been determined.
Now the next step is really tricky to visualize, but you can prove that triangles ACE and CGA are ASA-congruent (the angle measures provided in the image are there to help you out).
This means the corresponding sides CE and AG are equal in length (even though its not obvious in the image as it's not drawn to scale).
We also can know CF and AF are equal in length as they are the equal sides of the isosceles triangle ACF (take a look at the base angles if not convinced).
Therefore, CF - CE = AF - AG, which is the same as saying EF = GF.
Now as triangle DFG is equilateral, we know that DF = GF. Therefore, DF = EF. This means triangle DFE is isosceles with base angles DEF and EDF equal to each other.
We can know by looking at the shown angle measures and doing some basic calculation that angle AEF = 30. Therefore, DEF = 30 + x. As it is equal to half the sum of the measures of both base angles DEF and EDF, with the sum being equal to 100 (180 - 80), then the measure of DEF is 50.
Therefore 30 + x = 50 => x = 20
And we finally get our answer.
I love challenges, but this was way too messy for me.
https://image.ibb.co/hWvF2f/Stage-4.png
I had to restore the measures of some of the original angles (angles CAE and EAF) to clarify things here, and hopefully it's clear what angles correspond to which regions as indicated by the matches in colors. If still confusing, let me know. This solution is a bit of a mess, but I've tried to make this as clean as possible.
In this stage, one more line had to be drawn, a line from C to G which just happens to bisect angle ACB (remember that its measure is 20). If not convinced, work out how to prove triangles CAG and CBG are SSS-congruent by remembering that ABG is an equilateral triangle. Should be easy.
Consequently, all new angles in the image above should be easy to figure out on your own, so I won't spend too much energy trying to show how their measures have been determined.
Now the next step is really tricky to visualize, but you can prove that triangles ACE and CGA are ASA-congruent (the angle measures provided in the image are there to help you out).
This means the corresponding sides CE and AG are equal in length (even though its not obvious in the image as it's not drawn to scale).
We also can know CF and AF are equal in length as they are the equal sides of the isosceles triangle ACF (take a look at the base angles if not convinced).
Therefore, CF - CE = AF - AG, which is the same as saying EF = GF.
Now as triangle DFG is equilateral, we know that DF = GF. Therefore, DF = EF. This means triangle DFE is isosceles with base angles DEF and EDF equal to each other.
We can know by looking at the shown angle measures and doing some basic calculation that angle AEF = 30. Therefore, DEF = 30 + x. As it is equal to half the sum of the measures of both base angles DEF and EDF, with the sum being equal to 100 (180 - 80), then the measure of DEF is 50.
Therefore 30 + x = 50 => x = 20
And we finally get our answer.
I love challenges, but this was way too messy for me.
