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Aravind Reddy · Answer 1 · 2023-04-25T06:04:02+0000

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:

$\begin{gathered} 6x+37=(19)/(2)x-(31)/(2) \\ \\ 6x-(19)/(2)x=-(31)/(2)-37 \\ \\ (12x-19x)/(2)=(-31-74)/(2) \\ \\ -7x=-105 \\ \\ x=(-105)/(-7) \\ \\ x=15 \end{gathered}$

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

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