Question

quadrilateral rstq is a parallelogram. which of the following relationships must be true? ∠s ≅ ∠q and \overline{tq} ≅ \overline{sr} ∠t ≅ ∠r and \overline{tq} ≅ \overline{ts} ∠r ≅ ∠q and \overline{tq} ≅ \overline{sr} ∠s ≅ ∠q and \overline{rs} ≅ \overline{st}

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are congruent and opposite sides are congruent.

Step2: Analyze each option

• Option 1:

In parallelogram \(RSTQ\), \(\angle S\) and \(\angle Q\) are opposite angles (\(\angle S\cong\angle Q\)), and \(\overline{TQ}\) and \(\overline{SR}\) are opposite sides (\(\overline{TQ}\cong\overline{SR}\)).

• Option 2:

\(\angle T\) and \(\angle R\) are consecutive angles (not necessarily congruent, \(\angle T+\angle R = 180^{\circ}\) if \(ST\parallel RQ\)), and \(\overline{TQ}\) and \(\overline{TS}\) are adjacent sides (not necessarily congruent).

• Option 3:

\(\angle R\) and \(\angle Q\) are consecutive angles (not necessarily congruent, \(\angle R+\angle Q=180^{\circ}\) if \(RQ\parallel ST\)).

• Option 4:

\(\overline{RS}\) and \(\overline{ST}\) are adjacent sides (not necessarily congruent).

Answer:

\(\angle S\cong\angle Q\) and \(\overline{TQ}\cong\overline{SR}\) (the first option)