given: \\(\\overline{df} \\parallel \\overline{hg}\\), \\(f\\) is the midpoint of \\(\\overline{eg}\\)\\(\\ov…

Explanation:

Step1: Fill Statement 1

Given is (\overline{DF} \parallel \overline{HG}), so statement 1 is (\overline{DF} \parallel \overline{HG}).

Step2: Reason for Statement 2

Statement 2 says (F) is the midpoint of (\overline{EG}), and the reason is "Given" (since it's part of the given information).

Step3: Fill Statement 3

Given is (\overline{DF} \cong \overline{HG}), so statement 3 is (\overline{DF} \cong \overline{HG}).

Step4: Find Statement 4

Since (\overline{DF} \parallel \overline{HG}), the corresponding angles (like (\angle DFE) and (\angle HGF) or (\angle EDF) and (\angle EHG), depending on the transversal) are congruent. So statement 4 is (\angle DFE \cong \angle HGF) (or appropriate corresponding angles).

Step5: Reason for Statement 5

(\overline{EF} \cong \overline{FG}) because (F) is the midpoint of (\overline{EG}), and the definition of a midpoint states that it divides the segment into two congruent parts. So the reason is "Definition of Midpoint".

Step6: Reason for Statement 6

To prove (\triangle DEF \cong \triangle HFG), we have (\overline{DF} \cong \overline{HG}) (statement 3), (\angle DFE \cong \angle HGF) (statement 4), and (\overline{EF} \cong \overline{FG}) (statement 5). This satisfies the SAS (Side - Angle - Side) congruence criterion. So the reason is "SAS Congruence Criterion".

Answer:

1. (\boldsymbol{\overline{DF} \parallel \overline{HG}})
2. Given
3. (\boldsymbol{\overline{DF} \cong \overline{HG}})
4. (\boldsymbol{\angle DFE \cong \angle HGF}) (or appropriate corresponding angles)
5. Definition of Midpoint
6. SAS Congruence Criterion