Area Of A Kite.

In $\triangle{EHF}$ , $$ $\overline{\rm EH} \cong \overline{\rm HF} \implies$

$\overline{\rm HI}$ is the $\perp$ bisector of $EF\ \implies \angle{HIE}$ is a right angle and $\overline{\rm EI} \cong \overline{\rm IF}$

and $y = \overline{\rm HG} \cong \overline{\rm AC} \cong \overline{\rm EF} \implies \triangle{EHF}$ is an equilateral triangle $\implies m\angle{EHG} = 30^\circ \implies$

$m\angle{EGH} = m\angle{HEG} = 75^{\circ}$ in isosceles $\triangle{EHG} \implies m\angle{EGF} = \boxed{150^\circ}$

In isosceles $\triangle{EGH}$ the height $h = \sin(75^\circ)y = \cos(15^\circ)y$ and using the law of cosines

$\implies z = \overline{EG} = 2y\sin(15^\circ) \implies A_{\triangle{HEG}} = \dfrac{1}{2}\sin(30^{\circ}) = \dfrac{1}{4}y^2 \implies$

$A_{EGFH} = 2A_{\triangle{HEG}} = \dfrac{1}{2}y^2 = 50 \implies y = \boxed{10}$

$\implies m\angle{EGF} + y = 150 + 10 = \boxed{160}$ .