Question

statements

1. $overline{ab}congoverline{fg}$
2. $overleftrightarrow{bf}$ bisects $overline{ac}$ and $overline{dg}$.

3.
4.

1. $overline{bc}congoverline{df}$

reasons

1. given
2. given
3. definition of segment bisector
4. transitive property of segment congruence

5.

Explanation:

Step1: Apply segment - bisector definition

Since $\overleftrightarrow{BF}$ bisects $\overline{AC}$ and $\overline{DG}$, we have $\overline{AB}\cong\overline{BC}$ and $\overline{FG}\cong\overline{DF}$ by the definition of segment bisector.

Step2: Use transitive property

We know that $\overline{AB}\cong\overline{FG}$ (given). Also, $\overline{AB}\cong\overline{BC}$ and $\overline{FG}\cong\overline{DF}$. By the transitive property of segment congruence, if $\overline{AB}\cong\overline{FG}$, $\overline{AB}\cong\overline{BC}$, and $\overline{FG}\cong\overline{DF}$, then $\overline{BC}\cong\overline{DF}$.

Step3: State the reason for the last step

The reason for $\overline{BC}\cong\overline{DF}$ is the transitive property of segment congruence.

Answer:

1. $\overline{AB}\cong\overline{BC}$ and $\overline{FG}\cong\overline{DF}$; 4. Since $\overline{AB}\cong\overline{FG}$, $\overline{AB}\cong\overline{BC}$, $\overline{FG}\cong\overline{DF}$, then $\overline{BC}\cong\overline{DF}$; 5. Transitive property of segment congruence.