Question

prove: abcd is a parallelogram.

abcd is a rhombus.

statements

1. abcd is a rhombus.
2. \\(\overline{bc} \cong \overline{ad}\\) and \\(\overline{ab} \cong \overline{cd}\\)
3. \\(\overline{ac} \cong \overline{ac}\\)
4. \\(\delta abc \cong \delta cda\\)
5. \\(\angle 1 \cong \angle 4\\) and \\(\angle 2 \cong \angle 3\\)
6. \\(\overline{bc} \parallel \overline{ad}\\) and
7. abcd is a parallelogram.

reasons

1. given
2. def. of rhombus

3.
4.

1. cpctc
2. conv. of the alt. int. \\(\angle\\)s thrm.
3. def. \\(\parallelogram\\)

a. \\(\overline{bc} \parallel \overline{cd}\\)
b. symmetric
c. reflexive
d. sss
e. \\(\overline{ab} \parallel \overline{cd}\\)
f. sas

Explanation:

For Reason 3:

Step1: Recall congruence properties

The reflexive property of congruence states that any segment is congruent to itself. Here, $\overline{AC}$ is congruent to itself, so the reason for $\overline{AC} \cong \overline{AC}$ is the reflexive property. So the correct option is c. Reflexive.

Step2: For $\triangle ABC \cong \triangle CDA$

We have $\overline{BC} \cong \overline{AD}$, $\overline{AB} \cong \overline{CD}$ (from step 2) and $\overline{AC} \cong \overline{AC}$ (from step 3). By the SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So the reason for $\triangle ABC \cong \triangle CDA$ is SSS, so the correct option is d. SSS.

Step3: For the parallel line in step 6

We know that $\angle 1\cong\angle 4$ and $\angle 2\cong\angle 3$ (from step 5, CPCTC). By the converse of the alternate interior angles theorem, if alternate interior angles are congruent, then the lines are parallel. We already have $\overline{BC}\parallel\overline{AD}$ (from alternate interior angles $\angle 2$ and $\angle 3$). For the other pair of sides, since $\angle 1\cong\angle 4$, the lines $\overline{AB}$ and $\overline{CD}$ are parallel (because $\angle 1$ and $\angle 4$ are alternate interior angles formed by transversal $\overline{AC}$ with $\overline{AB}$ and $\overline{CD}$). So the missing parallel line is $\overline{AB}\parallel\overline{CD}$, so the correct option is e. $\overline{AB}\parallel\overline{CD}$

Answer:

s:

• Reason 3: c. Reflexive
• Reason 4: d. SSS
• Step 6: e. $\overline{AB}\parallel\overline{CD}$