Question

given: \\(\overline{df} \parallel \overline{hg}\\), \\(f\\) is the midpoint of \\(\overline{eg}\\)\\(\overline{df} \cong \overline{hg}\\)prove: \\(\triangle def \cong \triangle hfg\\)\

statements	reasons	\
---	---	\
1.	1. given	\
2. \\(f\\) is the midpoint of \\(eg\\)	2.	\
3.	3. given	\
4.	4. corresponding angles	\
5. \\(\overline{ef} \cong \overline{fg}\\)	5.	\
6. \\(\triangle def \cong \triangle hfg\\)	6.

Explanation:

Step1: Fill Statement 1

The first given is $\overline{DF} \parallel \overline{HG}$, $F$ is the midpoint of $\overline{EG}$ and $\overline{DF} \cong \overline{HG}$. So statement 1 should be $\overline{DF} \parallel \overline{HG}$, $F$ is the midpoint of $\overline{EG}$ (since reason is Given).

Step2: Reason for Statement 2

Statement 2: $F$ is the midpoint of $EG$ has reason "Given" (from the problem's given information: "Given: $\overline{DF} \parallel \overline{HG}$, $F$ is the midpoint of $\overline{EG}$...").

Step3: Fill Statement 3

The third given is $\overline{DF} \cong \overline{HG}$, so statement 3 is $\overline{DF} \cong \overline{HG}$ (reason is Given).

Step4: Find Corresponding Angles

Since $\overline{DF} \parallel \overline{HG}$, the corresponding angles are $\angle DFE \cong \angle HFG$ (alternate interior angles, but the reason is given as Corresponding Angles, so statement 4 is $\angle DFE \cong \angle HFG$.

Step5: Reason for Statement 5

$\overline{EF} \cong \overline{FG}$ because $F$ is the midpoint of $EG$, and by definition of midpoint, a midpoint divides a segment into two congruent segments. So reason for statement 5 is "Definition of Midpoint".

Step6: Prove Triangle Congruence

We have $\overline{DF} \cong \overline{HG}$ (statement 3), $\angle DFE \cong \angle HFG$ (statement 4), $\overline{EF} \cong \overline{FG}$ (statement 5). So by SAS (Side - Angle - Side) congruence criterion, $\triangle DEF \cong \triangle HFG$. So reason for statement 6 is "SAS Congruence Criterion".

Filling the table:

Statements	Reasons
2. $F$ is the midpoint of $EG$	2. Given
3. $\overline{DF} \cong \overline{HG}$	3. Given
4. $\angle DFE \cong \angle HFG$	4. Corresponding Angles (Alternate Interior Angles as $DF\parallel HG$)
5. $\overline{EF} \cong \overline{FG}$	5. Definition of Midpoint
6. $\triangle DEF \cong \triangle HFG$	6. SAS Congruence Criterion

Answer:

The completed two - column proof is as shown above. The key steps are using given information, properties of midpoints, corresponding angles from parallel lines, and SAS congruence to prove the triangles congruent.