Question

quadrilateral rstq is a parallelogram. which of the following relationships must be true? ∠s ≅ ∠r, tq ≅ qr, rs ≅ rq, ∠t ≅ ∠r

Explanation:

Step1: Recall the properties of a parallelogram

In a parallelogram, opposite angles are congruent.

Step2: Analyze each option

• For \(\angle S\cong\angle R\): In a parallelogram \(RSTQ\), \(\angle S\) and \(\angle R\) are consecutive angles. Consecutive angles in a parallelogram are supplementary (\(\angle S+\angle R = 180^{\circ}\) unless it is a rectangle), not necessarily congruent.
• For \(\overline{TQ}\cong\overline{QR}\): In a parallelogram, opposite sides are congruent (\(\overline{TQ}\cong\overline{RS}\) and \(\overline{TR}\cong\overline{SQ}\)). There is no property that \(\overline{TQ}\cong\overline{QR}\) in a general parallelogram.
• For \(\overline{RS}\cong\overline{RQ}\): In a parallelogram, opposite sides are congruent (\(\overline{TQ}\cong\overline{RS}\) and \(\overline{TR}\cong\overline{SQ}\)). There is no property that \(\overline{RS}\cong\overline{RQ}\) in a general parallelogram.
• For \(\angle T\cong\angle R\): Since \(RSTQ\) is a parallelogram, by the property of parallelograms (opposite angles are congruent), \(\angle T\) and \(\angle R\) are opposite angles. So \(\angle T\cong\angle R\) must be true.

Answer:

\(\angle T\cong\angle R\)