Question

step 1: statement - abcd is a rhombus, \\(\overline{df}\\) bisects \\(\overline{bc}\\); reason - given. step 2: statement - \\(\overline{be} \cong \overline{ce}\\); reason - a segment bisector divides a segment into two congruent segments. step 3: statement - \\(\angle dec \cong \angle feb\\); reason - vertical angles are congruent. step 4: statement - \\(\overline{dc} \parallel \overline{ab}\\); reason - opposite sides of a parallelogram / rhombus / rectangle / square are parallel. step 5: statement - \\(\angle c \cong \angle ebf\\); reason - select a reason. (image of rhombus abcd with df bisecting bc, e as intersection point, and ab, dc with arrows indicating parallelism)

Explanation:

Step1: Recall properties of parallel lines

Since \( \overline{DC} \parallel \overline{AB} \) (from Step 4), and \( \overline{BC} \) is a transversal cutting these parallel lines.

Step2: Identify the angle relationship

When two parallel lines are cut by a transversal, alternate interior angles are congruent. Here, \( \angle C \) and \( \angle EBF \) are alternate interior angles formed by transversal \( \overline{BC} \) intersecting \( \overline{DC} \) and \( \overline{AB} \). So the reason is "Alternate Interior Angles Theorem (If two parallel lines are cut by a transversal, then alternate interior angles are congruent)".

Answer:

The reason for \( \angle C \cong \angle EBF \) is "Alternate Interior Angles are congruent (when two parallel lines are cut by a transversal)".