#### Answer

1) Prove $\triangle FED\cong\triangle GED$, using method AAS:
- Prove $\angle EDF\cong\angle EDG$
- Show by given that $\angle F\cong\angle G$
- Show by identity that $\overline{DE}\cong\overline{DE}$
2) Then $\overline{FE}\cong\overline{GE}$ by CPCTC
3) Then $E$ is the midpoint of $\overline{FG}$

#### Work Step by Step

*PLANNING:
First, we need to prove $\triangle FED\cong\triangle GED$.
Then we can deduce $\overline{FE}\cong\overline{GE}$. So, $E$ is the midpoint of $\overline{FG}$.
1. $\vec{DE}$ bisects $\angle FDG$ (Given)
2. $\angle EDF\cong\angle EDG$ (The bisector of an angle divides it into 2 congruent angles)
3. $\angle F\cong\angle G$ (Given)
4. $\overline{DE}\cong\overline{DE}$ (Identity)
So now we have 2 angles and a non-included side of $\triangle FED$ are congruent with 2 corresponding angles and a non-included side of $\triangle GED$. Therefore,
3. $\triangle FED\cong\triangle GED$ (AAS)
4. $\overline{FE}\cong\overline{GE}$ (CPCTC)
5. $E$ is the midpoint of $\overline{FG}$ (if a point divides a line into 2 congruent lines, then that point is the midpoint of that line)