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\tstatement\treason
\tabcd is a parallelogram\t
1\t\\(\overline{fc} \cong \overline{ec}\\)\tgiven
\t\\(\overline{de} \perp \overline{bc}\\)\t
\t\\(\overline{bf} \perp \overline{dc}\\)\t
2\t\\(\angle dec\\) is a right angle\tperpendicular lines form right angles
3\t\\(\angle bfc\\) is a right angle\tan angle congruent to a right angle is a right angle
4\t\\(\angle dec \cong \angle bfc\\)\tparallel lines cut by a transversal form congruent alternate interior angles
5\t\\(\angle c \cong \angle c\\)\treflexive property
6\t\\(\triangle dec \cong \triangle bfc\\)\taas
7\t\\(\overline{dc} \cong \overline{bc}\\)\tall sides of a rhombus / square are congruent
8\tabcd is a rhombus\tabcd is a rhombus because it is a quadrilateral with one pair of opposite sides parallel

Explanation:

Step1: Analyze Step 7's Reason

The reason for \(\overline{DC}\cong\overline{BC}\) in Step 7 is incorrect. Since \(\triangle DEC\cong\triangle BFC\) (AAS from Step 6), corresponding parts of congruent triangles are congruent (CPCTC), so \(\overline{DC}\cong\overline{BC}\) should use CPCTC, not "All sides of a rhombus/square are congruent" (that's a property of rhombus, not a reason to prove sides congruent here).

Step2: Analyze Step 8's Reason

To prove \(ABCD\) is a rhombus, we know \(ABCD\) is a parallelogram (given). A parallelogram with a pair of adjacent sides congruent (\(\overline{DC}\cong\overline{BC}\)) is a rhombus (definition of rhombus: parallelogram with all sides equal, or adjacent sides equal). The given reason "one pair of opposite sides parallel" is incorrect because a parallelogram already has both pairs of opposite sides parallel. The correct reason is that a parallelogram with adjacent sides congruent is a rhombus (since in parallelogram \(AB\parallel DC\), \(AD\parallel BC\), and if \(DC = BC\), then all sides are equal as opposite sides of parallelogram are equal).

Answer:

• Step 7 Reason: Should be "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)" (because \(\triangle DEC\cong\triangle BFC\), so \(DC = BC\) by CPCTC).
• Step 8 Reason: Should be "a parallelogram with a pair of adjacent sides congruent" (or "a parallelogram with all sides congruent", since in parallelogram \(ABCD\), \(AB = DC\), \(AD = BC\), and \(DC = BC\) implies \(AB = BC = CD = DA\)).