SOLUTION: OC bisects AOB, OD bisects AOC OE bisects AOD, DF bisects AOE and OG busects FOC. If m BOF=120, then m DOE=? If m COG=35, then m EOG=? Draw a diagram and show all work.

You made a mistake and typed "DF bisects AOE" when you meant 
"OF bisects AOE".  I hope you didn't make any more mistakes
with any of those letters.  I'm assuming you typed all the
others correct. I wondered about "OG busects FOC" since that
has nothing to do with the first problem, so I'm not even
going to draw in OG for the first problem.  So if you made a
mistake there too, other than to misspell "bisect" as "busect",
then this is different from the problem you wanted to solve. 
But maybe it is the same one, or maybe it is similar enough.

Since OF bisects AOE, we let AOF and FOE be x degrees each.
Since OE bisects AOD, and since AOE is 2x, then DOE is also 2x.
Since OD bisects AOC, and since AOD is 4x, then DOC is also 4x.
Since OC bisects AOB, and since AOC is 8x, then BOC is also 8x.
The red arc indicates angle BOF which is given as 120°.




The problem asks for DOE, indicated by the green arc.





We add up the parts of the 120° angle BOF and we have 

EOF + DOE + DOC + BOC = 120°
 x  + 2x  + 4x  +  8x = 120 
                  15x = 120
                    x = 8°

The problem asks for DOE, indicated by the green arc, which
is 2x so the answer, assuming you typed everything else OK,
the answer is 2x = 2(8°) = 16°.

Now you said "OG busects FOC", which has nothing to do with 
the first problem, assuming you made no other mistakes in
typing all those letters. I'll draw in OG in green bisecting 
FOC, based on what you typed to show that it has nothing to 
do with the first problem, assuming you made no other typos.



Now your second problem states:

If m COG=35, then m EOG=?

Well, based on what you have typed, COG is not 35, it's 28.

So you've botched the problem.  Please don't do that to us
again.

Be very careful not to mistype anything. Read and re-read and 
re-read everything.

Edwin