Question

statement reason
abcd is a rhombus given
\\(\overline{df}\\) bisects \\(\overline{bc}\\)
\\(\overline{be} \cong \overline{ce}\\) a segment bisector divides a segment into two congruent segments
\\(\angle dec \cong \angle feb\\) vertical angles are congruent
\\(\overline{dc} \parallel \overline{ab}\\) opposite sides of a parallelogram / rhombus / rectangle / square are parallel
\\(\angle c \cong \angle ebf\\) parallel lines cut by a transversal form congruent alternate interior angles
\\(\triangle dec \cong \triangle feb\\) asa
\\(\overline{bf} \cong \overline{dc}\\)
image of a rhombus abcd with df bisecting bc, intersecting at e, and f on the extension of ab

Explanation:

Step1: Analyze the proof structure

We have a rhombus \(ABCD\), \(DF\) bisects \(BC\) (so \(BE\cong CE\) by segment bisector definition), \(\angle DEC\cong\angle FEB\) (vertical angles), \(DC\parallel AB\) (opposite sides of rhombus), \(\angle C\cong\angle EBF\) (alternate interior angles from parallel lines), then \(\triangle DEC\cong\triangle FEB\) by ASA. For \(\overline{BF}\cong\overline{DC}\), we use the property of congruent triangles.

Step2: Determine the reason for \(\overline{BF}\cong\overline{DC}\)

In congruent triangles, corresponding parts are congruent (CPCTC). Since \(\triangle DEC\cong\triangle FEB\), their corresponding sides \(\overline{BF}\) (corresponding to \(\overline{DC}\)) are congruent. So the reason is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Answer:

The reason for \(\overline{BF}\cong\overline{DC}\) is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".