Question

given: abcd is a parallelogram. prove: \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{bc} \cong \overline{da}\\) assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Draw diagonal \( AC \)

Draw \( \overline{AC} \) to divide parallelogram \( ABCD \) into \( \triangle ABC \) and \( \triangle CDA \).

Step2: Identify parallel sides

In parallelogram \( ABCD \), \( \overline{AB} \parallel \overline{CD} \) and \( \overline{BC} \parallel \overline{DA} \) (Definition of parallelogram).

Step3: Find alternate interior angles

• \( \angle BAC \cong \angle DCA \) (Alternate Interior Angles Theorem, since \( \overline{AB} \parallel \overline{CD} \) and \( \overline{AC} \) is transversal).
• \( \angle BCA \cong \angle DAC \) (Alternate Interior Angles Theorem, since \( \overline{BC} \parallel \overline{DA} \) and \( \overline{AC} \) is transversal).

Step4: Identify common side

\( \overline{AC} \cong \overline{AC} \) (Reflexive Property of Congruence).

Step5: Prove triangles congruent

\( \triangle ABC \cong \triangle CDA \) (ASA Congruence Criterion, \( \angle BAC \cong \angle DCA \), \( \overline{AC} \cong \overline{AC} \), \( \angle BCA \cong \angle DAC \)).

Step6: Corresponding parts of congruent triangles

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

Answer:

By drawing diagonal \( AC \), using properties of parallelogram (parallel sides), Alternate Interior Angles Theorem, Reflexive Property, ASA congruence for \( \triangle ABC \) and \( \triangle CDA \), and CPCTC, we prove \( \overline{AB} \cong \overline{CD} \) and \( \overline{BC} \cong \overline{DA} \).