Angle DEF is a inscribed angle.
A inscribed angle (DEF) is equal to the half of the arc that the two sides cout out of the circle (Arc FGD):
![\begin{gathered} m\angle DEF=(1)/(2)mFGD \\ \\ m\angle DEF=6x+37 \\ \text{mFGD}=19x-31 \\ \\ 6x+37=(1)/(2)(19x-31) \end{gathered}]()
Use the equation above to solve x:
Use the value of x to find the measure of angle DEF:
![\begin{gathered} m\angle DEF=6x+37 \\ m\angle DEF=6(15)+37 \\ m\angle DEF=90+37 \\ m\angle DEF=127 \end{gathered}]()
Then, the measure of angle DEF is 127º