Question

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

1. abcd is a parallelogram\
2. draw \\(\overline{ac}\\)\

reasons\

1. given\
2. unique line postulate\

diagram of parallelogram abcd with diagonal ac\
options: angles, segments, triangles, statements, reasons; with choices like reflexive property, def. of alt. interior angles, def. of parallelogram, alternate interior angles theorem

Explanation:

Step1: Recall parallelogram properties

In a parallelogram \(ABCD\), \(AB \parallel CD\) and \(BC \parallel DA\) (def. of parallelogram). After drawing \(AC\), we have \(\angle BAC \cong \angle DCA\) and \(\angle BCA \cong \angle DAC\) (alternate interior angles theorem, since \(AB \parallel CD\) and \(BC \parallel DA\) with transversal \(AC\)). Also, \(AC \cong AC\) (reflexive property).

Step2: Prove triangle congruence

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

Step3: Corresponding parts of congruent triangles

Since \(\triangle ABC \cong \triangle CDA\), their corresponding sides are congruent. So, \(AB \cong CD\) and \(BC \cong DA\) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

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

1. Draw \( \overline{AC} \) (unique line postulate, as two points \(A\) and \(C\) determine a unique line).
2. \(AB \parallel CD\) and \(BC \parallel DA\) (definition of a parallelogram).
3. \(\angle BAC \cong \angle DCA\), \(\angle BCA \cong \angle DAC\) (alternate interior angles theorem, since \(AB \parallel CD\) and \(BC \parallel DA\) with transversal \(AC\)).
4. \(AC \cong AC\) (reflexive property of congruence).
5. \(\triangle ABC \cong \triangle CDA\) (ASA congruence criterion: \(\angle BAC \cong \angle DCA\), \(AC \cong AC\), \(\angle BCA \cong \angle DAC\)).
6. \(AB \cong CD\), \(BC \cong DA\) (CPCTC: Corresponding Parts of Congruent Triangles are Congruent).

Thus, \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{DA} \) are proven.