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 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 reasons: 1. given 2. unique line postulate 3. def. of alt. interior angles 4. def. of alt. interior angles

Explanation:

Step1: Recall parallelogram properties

In parallelogram \(ABCD\), \(AB \parallel CD\) and \(BC \parallel DA\) (def. of parallelogram). Draw diagonal \(AC\).

Step2: Identify alternate interior angles

For \(AB \parallel CD\) and transversal \(AC\), \(\angle BAC \cong \angle DCA\) (alt. interior angles). For \(BC \parallel DA\) and transversal \(AC\), \(\angle BCA \cong \angle DAC\) (alt. interior angles).

Step3: Prove triangle congruence

\(AC = AC\) (reflexive property). By ASA, \(\triangle ABC \cong \triangle CDA\).

Step4: Corresponding parts of congruent triangles

Since \(\triangle ABC \cong \triangle CDA\), \(AB \cong CD\) and \(BC \cong DA\) (CPCTC).

Answer:

By proving \(\triangle ABC \cong \triangle CDA\) (ASA) using parallelogram properties and alternate interior angles, we conclude \(AB \cong CD\) and \(BC \cong DA\).