Complete the proof of the Converse of the Triangle Proportionality Theorem.

Steps to Solve

1. Fill in for Statement 2
   The angles $\angle GEH$ and $\angle DEF$ are corresponding angles formed by the transversal $DE$ intersecting the parallel lines $DF$ and $GH$.
   Thus, we can use the property of corresponding angles.
   Reason: Corresponding Angles Postulate.

2. Fill in for Statement 3
   Since $\angle GEH$ and $\angle DEF$ are equal, we can add 1 (which represents an equality of angles) to the existing equation to keep the comparison valid.
   Reason: Addition Property of Equality.

3. Fill in for Statement 6
   Here, we state that the segments $FE$ and $DE$ are equal lengths due to equality established in the previous steps involving proportions.
   Reason: Reflexive Property of Congruence.

4. Fill in for Statement 8
   Triangles $\triangle FDE$ and $\triangle HGE$ are similar by the Angle-Angle (AA) criterion for triangle similarity, given that two angles are equal.
   Reason: Side-Angle-Side Similarity Theorem.

5. Fill in for Statement 10
   By the properties of proportion and the similarity established between the two triangles, we conclude that $DF$ is parallel to $GH$.
   Reason: Converse of the Triangle Proportionality Theorem.