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):
[tex]\begin{gathered} m\angle DEF=\frac{1}{2}mFGD \\ \\ m\angle DEF=6x+37 \\ \text{mFGD}=19x-31 \\ \\ 6x+37=\frac{1}{2}(19x-31) \end{gathered}[/tex]
Use the equation above to solve x:
[tex]\begin{gathered} 6x+37=\frac{19}{2}x-\frac{31}{2} \\ \\ 6x-\frac{19}{2}x=-\frac{31}{2}-37 \\ \\ \frac{12x-19x}{2}=\frac{-31-74}{2} \\ \\ -7x=-105 \\ \\ x=\frac{-105}{-7} \\ \\ x=15 \end{gathered}[/tex]
Use the value of x to find the measure of angle DEF:
[tex]\begin{gathered} m\angle DEF=6x+37 \\ m\angle DEF=6(15)+37 \\ m\angle DEF=90+37 \\ m\angle DEF=127 \end{gathered}[/tex]Then, the measure of angle DEF is 127º
