Question

given: abcd is a parallelogram, \\(\overline{fc} \cong \overline{ec}\\), \\(\overline{de} \perp \overline{bc}\\) and \\(\overline{bf} \perp \overline{dc}\\).
prove: abcd is a rhombus.

step | statement | reason
--- | --- | ---
1 | abcd is a parallelogram; \\(\overline{fc} \cong \overline{ec}\\); \\(\overline{de} \perp \overline{bc}\\); \\(\overline{bf} \perp \overline{dc}\\) | given
2 | \\(\angle dec\\) is a right angle | perpendicular lines form right angles
3 | \\(\angle bfc\\) is a right angle | perpendicular lines form right angles
4 | \\(\angle dec \cong \angle bfc\\) | all right angles are congruent
5 | \\(\angle c \cong \angle c\\) | reflexive property
6 | \\(\triangle dec \cong \triangle bfc\\) | asa
7 | \\(dc \cong bc\\) | corresponding parts of congruent triangles are congruent (cpctc)
8 | abcd is a rhombus | abcd is a rhombus because it is a quadrilateral with four congruent sides (dropdown options: one pair of opposite sides parallel, one pair of opposite sides congruent, two pairs of opposite sides parallel, two pairs of opposite sides congruent, four congruent sides)

Explanation:

Step1: Confirm given statements

$ABCD \text{ is a parallelogram}, \overline{FC} \cong \overline{EC}, \overline{DE} \perp \overline{BC}, \overline{BF} \perp \overline{DC}$

Step2: Identify right angles

$\angle DEC = 90^\circ, \angle BFC = 90^\circ$ (Perpendicular lines form right angles)

Step3: Congruent right angles

$\angle DEC \cong \angle BFC$ (All right angles are congruent)

Step4: Reflexive angle congruence

$\angle C \cong \angle C$ (Reflexive Property)

Step5: Prove triangle congruence

$\triangle DEC \cong \triangle BFC$ (ASA Congruence)

Step6: CPCTC for side congruence

$\overline{DC} \cong \overline{BC}$ (Corresponding Parts of Congruent Triangles are Congruent)

Step7: Parallelogram to rhombus

In parallelogram $ABCD$, adjacent sides $\overline{DC} \cong \overline{BC}$, so all sides are congruent.

Answer:

The completed proof shows $ABCD$ is a rhombus, as it is a parallelogram with a pair of congruent adjacent sides (which implies all four sides are congruent, satisfying the definition of a rhombus). The correct reason for Step 8 is: $ABCD$ is a rhombus because it is a parallelogram with a pair of congruent adjacent sides (or equivalently, a parallelogram with four congruent sides).