Question

given: abcd is a parallelogram. prove: \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{bc} \cong \overline{da}\\) diagram of parallelogram abcd with diagonal ac select the correct reason for this statement. statements: 1. abcd is a parallelogram 2. draw \\(\overline{ac}\\) 3. \\(\angle bca\\) and \\(\angle dac\\) are alt. interior angles 4. \\(\angle dca\\) and \\(\angle bac\\) are alt. interior angles 5. \\(\overline{ab} \parallel \overline{cd}\\) 6. \\(\overline{ac} \cong \overline{ac}\\) reasons: 1. given 2. unique line postulate 3. def. of alt. interior angles 4. def. of alt. interior angles 5. def. of parallelogram 6. reflexive property reasons options: asa, cpctc, def. of parallelogram, alternate interior angles theorem; statements boxes: bc, ||, da

Explanation:

To solve the proof for the parallelogram \(ABCD\) and show \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{DA} \), we follow these steps:

Step 1: Given Information

We know \(ABCD\) is a parallelogram (given).

Step 2: Draw Diagonal \( \overline{AC} \)

By the unique line postulate, we can draw diagonal \( \overline{AC} \) (this is a valid construction to create two triangles \( \triangle ABC \) and \( \triangle CDA \)).

Step 3: Alternate Interior Angles

Since \(ABCD\) is a parallelogram, \( \overline{AB} \parallel \overline{CD} \) and \( \overline{BC} \parallel \overline{DA} \) (definition of a parallelogram).

• For \( \overline{AB} \parallel \overline{CD} \) and transversal \( \overline{AC} \), \( \angle BAC \) and \( \angle DCA \) are alternate interior angles (so \( \angle BAC \cong \angle DCA \) by the alternate interior angles theorem).
• For \( \overline{BC} \parallel \overline{DA} \) and transversal \( \overline{AC} \), \( \angle BCA \) and \( \angle DAC \) are alternate interior angles (so \( \angle BCA \cong \angle DAC \) by the alternate interior angles theorem).

Step 4: Reflexive Property of Congruence

\( \overline{AC} \cong \overline{AC} \) (reflexive property: a segment is congruent to itself).

Step 5: Prove Triangles Congruent

By the ASA (Angle-Side-Angle) congruence criterion:

• \( \angle BAC \cong \angle DCA \) (from Step 3),
• \( \overline{AC} \cong \overline{AC} \) (from Step 4),
• \( \angle BCA \cong \angle DAC \) (from Step 3).

Thus, \( \triangle ABC \cong \triangle CDA \) by ASA.

Step 6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle ABC \cong \triangle CDA \), their corresponding sides are congruent:

• \( \overline{AB} \cong \overline{CD} \) (corresponding sides),
• \( \overline{BC} \cong \overline{DA} \) (corresponding sides).

For the missing statement in the proof (the last row of the "Statements" column), we need to show the congruence of the sides. From the diagram and the proof, the missing statement should be \( \overline{BC} \cong \overline{DA} \) (matching the goal of proving \( \overline{BC} \cong \overline{DA} \)).

Answer:

\(\overline{BC} \cong \overline{DA}\) (with the reason being CPCTC, as the triangles are congruent and these are corresponding sides).