Question

step | statement | reason
1 | abcd is a parallelogram
e is the midpoint of \\(\overline{ab}\\)
f is the midpoint of \\(\overline{dc}\\) | given
2 | \\(\overline{ad} \cong \overline{cb}\\) | opposite sides of a parallelogram / rhombus / rectangle / square are congruent
3 | \\(\overline{dc} \cong \overline{ab}\\) | opposite sides of a parallelogram / rhombus / rectangle / square are congruent
4 | \\(fc = \frac{1}{2}dc\\) | when a midpoint divides a segment into two parts, each part is half the length of the whole segment
5 | \\(ae = \frac{1}{2}ab\\) | when a midpoint divides a segment into two parts, each part is half the length of the whole segment
6 | \\(\overline{fc} \cong \overline{ae}\\) | segments that are half the lengths of congruent segments are congruent
7 | \\(\angle a \cong \angle c\\) | opposite angles of a parallelogram / rhombus / rectangle / square are congruent
8 | \\(\triangle ade \cong \triangle cbf\\) | sas
9 | \\(\overline{de} \cong \overline{fb}\\) | select a reason...
image of parallelogram abcd with points e (midpoint of ab) and f (midpoint of dc), and triangles ade and cbf

Explanation:

Step1: Recall Congruent Triangles Property

If two triangles are congruent, their corresponding sides are congruent.
We know from step 8 that $\triangle ADE \cong \triangle CBF$ (by SAS).

Step2: Apply Corresponding Parts of Congruent Triangles

In congruent triangles $\triangle ADE$ and $\triangle CBF$, the sides $\overline{DE}$ and $\overline{FB}$ are corresponding sides. By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, if two triangles are congruent, then their corresponding sides are congruent. So, since $\triangle ADE \cong \triangle CBF$, we have $\overline{DE} \cong \overline{FB}$ because they are corresponding sides of congruent triangles.

Answer:

Corresponding Parts of Congruent Triangles are Congruent (CPCTC)