SOLUTION: point G lies in the interior of angle DEF. The ratio of angle GEF to angle GED is 3 to 5, and the measuurement pf DEF = 64°. Find the measurement of angle DEG and the measurement

It is strange formulation of the problem :


    - You call the same angle GEF and FEG by two different names;

    - you call the same angle GED and DEG by two different names.


Do you make it intentionally to transform a simple problem into more complicated ?


--------------


Solution


Since the ratio of two angles GEF to GED is 3:5, you can write

      m(GEF) = 3x   and  m(GED) = 5x.

where x is their common angular measure.


Then  3x + 5x = 8x = 64;  hence x = 64/8 = 8 degrees.


From it, there is only one step to the 


ANSWER.  Angle GEF measure is 3*8 = 24 degrees;  angle GED measure is 5*8 = 40 degrees.


CHECK.   = .    ! Correct !