Question

step 1 statement rstu is a parallelogram 2 \\(\overline{rs}\\) is parallel to \\(\overline{tu}\\) and \\(\overline{ru}\\) is parallel to \\(\overline{ts}\\) 3 \\(\angle rsu \cong \angle tus\\) and \\(\angle rus \cong \angle tsu\\) 4 \\(\overline{su} \cong \overline{us}\\) 5 \\(\triangle rsu \cong \triangle tus\\) 6 \\(\overline{rs} \cong \overline{tu}\\) and \\(\overline{ru} \cong \overline{ts}\\) select all reasons that support one or more statements in the proof. \\(\square\\) symmetric property \\(\square\\) definition of a parallelogram \\(\square\\) alternate interior angles are congruent \\(\square\\) opposite sides of a parallelogram are congruent \\(\square\\) corresponding parts of congruent triangles are congruent

Explanation:

1. Symmetric property: Step 4 states \(\overline{SU}\cong\overline{US}\), which is the symmetric property of congruence (if \(A\cong B\), then \(B\cong A\)).
2. Definition of a parallelogram: Step 2 follows from Step 1 using the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel).
3. Alternate interior angles are congruent: Step 3 (\(\angle RSU\cong\angle TUS\) and \(\angle RUS\cong\angle TSU\)) is true because \(RS\parallel TU\) and \(RU\parallel TS\) (from Step 2), so alternate interior angles formed by transversal \(SU\) are congruent.
4. Corresponding parts of congruent triangles are congruent (CPCTC): Step 6 (\(\overline{RS}\cong\overline{TU}\) and \(\overline{RU}\cong\overline{TS}\)) follows from Step 5 (\(\triangle RSU\cong\triangle TUS\)) by CPCTC.
5. The "opposite sides of a parallelogram are congruent" is the conclusion of Step 6, not a reason for the steps in the proof (the proof is deriving this property, so this is not a supporting reason here). Also, "definition of a parallelogram" supports Step 2, "symmetric property" supports Step 4, "alternate interior angles" supports Step 3, and "CPCTC" supports Step 6.

Answer:

• symmetric property
• definition of a parallelogram
• alternate interior angles are congruent
• corresponding parts of congruent triangles are congruent