Question

given: \\(\overline{eh}\\) bisects \\(\overline{dg}\\)\
\\(\overline{de} \parallel \overline{gh}\\)\
prove: \\(\overline{de} \cong \overline{gh}\\)\
step statement reason\
1 \\(\overline{eh}\\) bisects \\(\overline{dg}\\) given\
2 \\(\overline{de} \parallel \overline{gh}\\) given\
3 \\(\angle dfe \cong \angle gfh\\) vertical angles property\
4 \\(\boxed{}\\) reason?

Explanation:

Step1: Recall Bisect Definition

Since \( \overline{EH} \) bisects \( \overline{DG} \), \( DF = FG \) (definition of segment bisector).

Step2: Identify Alternate Interior Angles

Because \( \overline{DE} \parallel \overline{GH} \), \( \angle D \cong \angle G \) (Alternate Interior Angles Theorem).

Step3: Prove Triangles Congruent

We have \( \angle DFE \cong \angle GFH \) (Vertical Angles), \( DF = FG \), and \( \angle D \cong \angle G \). So \( \triangle DFE \cong \triangle GFH \) (ASA Congruence).

Step4: Corresponding Parts of Congruent Triangles

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \overline{DE} \cong \overline{GH} \). For step 4 in the proof table, the statement could be \( \triangle DFE \cong \triangle GFH \) with reason ASA, or directly \( \overline{DE} \cong \overline{GH} \) with reason CPCTC. Assuming we need to fill step 4 to lead to the proof, let's use the triangle congruence first. So step 4 statement: \( \triangle DFE \cong \triangle GFH \), reason: ASA (Angle - Side - Angle) Congruence (since \( \angle D \cong \angle G \), \( DF = FG \), \( \angle DFE \cong \angle GFH \)).

Answer:

For step 4, the statement is \( \triangle DFE \cong \triangle GFH \) and the reason is "ASA Congruence" (or if moving to the final proof, statement \( \overline{DE} \cong \overline{GH} \) with reason "CPCTC"). If following the proof steps, the key step 4 to prove triangle congruence is \( \triangle DFE \cong \triangle GFH \) by ASA.