## Elementary Geometry for College Students (6th Edition)

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 $\overline{FG}$
*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)