Answer
$m\angle EHG=93^{\circ}$
Work Step by Step
We are given that the $m\angle EOG=82^{\circ}$. Radii $\overline{OE}=\overline{OG}$, creating an isosceles triangle. The sum of the interior angles of a triangle is $180^{\circ}$. Subtracting the vertex angle, and dividing the result by two will give us the measure of the base angles...more importantly, the $m\angle E$.
$180-82=98\div2=49$
$m\angle E=49^{\circ}$
We learned from part d that the $m\angle DGE=38^{\circ}$. Using the sum of the interior angles for $\triangle GEH$...
$180-49-38=93$
$m\angle EHG=93^{\circ}$
