In the figure below, EFG is an isosceles triangle. DEG is a straight line and EG = FG. $\angle DEF = 111^\circ$ and $\angle HGE = 63^\circ$. Find $\angle y$. Answer: ___

1. Find $\angle FEG$ using angles on a straight line

$\angle FEG = 180^\circ - 111^\circ = 69^\circ$

2. Find $\angle EGF$ using the properties of the isosceles triangle

Since $EG = FG$, $\triangle EFG$ is an isosceles triangle with base angles $\angle FEG = \angle EFG = 69^\circ$.
$\angle EGF = 180^\circ - 69^\circ - 69^\circ = 42^\circ$

3. Find the interior $\angle HGF$

$\angle HGF = \angle HGE + \angle EGF = 63^\circ + 42^\circ = 105^\circ$

4. Find the reflex angle $y$

$\text{Reflex } \angle y = 360^\circ - 105^\circ = 255^\circ$