Question

given: abcd is a parallelogram. prove: \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{bc} \cong \overline{da}\\) correct! assemble the next 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

Explanation:

Step1: Identify triangles

We have parallelogram \(ABCD\) with diagonal \(AC\), so triangles \(ABC\) and \(CDA\) are formed.

Step2: List congruent parts

• \(AB \parallel CD\) (def. of parallelogram), so \(\angle BAC \cong \angle DCA\) (alt. interior angles).
• \(AD \parallel BC\) (def. of parallelogram), so \(\angle BCA \cong \angle DAC\) (alt. interior angles).
• \(AC \cong AC\) (reflexive property).

Step3: Prove triangle congruence

By ASA (Angle - Side - Angle) criterion, \(\triangle ABC \cong \triangle CDA\).

Step4: Use CPCTC

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(AB \cong CD\) and \(BC \cong DA\).

Answer:

To prove \(AB \cong CD\) and \(BC \cong DA\) in parallelogram \(ABCD\):

1. Draw diagonal \(AC\) (unique line postulate).
2. \(AB \parallel CD\) and \(AD \parallel BC\) (def. of parallelogram), so \(\angle BAC \cong \angle DCA\), \(\angle BCA \cong \angle DAC\) (alt. interior angles theorem).
3. \(AC \cong AC\) (reflexive property).
4. \(\triangle ABC \cong \triangle CDA\) (ASA).
5. \(AB \cong CD\), \(BC \cong DA\) (CPCTC).

The next statement after step 6 (\(AC \cong AC\)) would be proving \(\triangle ABC \cong \triangle CDA\) (using ASA) and then applying CPCTC to get the required congruences.