Question

topic: quadrilaterals
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
quadrilateral rstq is a parallelogram.
which of the following relationships must be true?
∠s≅∠q and (overline{rs}congoverline{st})
∠r≅∠q and (overline{tq}congoverline{sr})
∠t≅∠r and (overline{tq}congoverline{ts})
∠s≅∠q and (overline{tq}congoverline{sr})

Explanation:

Step1: Recall parallelogram properties

In a parallelogram, opposite - angles are congruent and opposite - sides are congruent.
For parallelogram \(RSTQ\), \(\angle S\) and \(\angle Q\) are opposite angles, so \(\angle S\cong\angle Q\). Also, \(\overline{TQ}\) and \(\overline{SR}\) are opposite sides, so \(\overline{TQ}\cong\overline{SR}\).

Step2: Analyze each option

• Option 1: \(\angle S\cong\angle Q\) is correct, but \(\overline{RS}\cong\overline{ST}\) is not a general property of parallelograms (adjacent sides are not necessarily congruent).
• Option 2: \(\angle R\) and \(\angle Q\) are adjacent angles, not opposite angles, so \(\angle R

ot\cong\angle Q\) in general.

• Option 3: \(\angle T\) and \(\angle R\) are adjacent angles, not opposite angles, so \(\angle T

ot\cong\angle R\) in general, and \(\overline{TQ}\) and \(\overline{TS}\) are adjacent sides, not opposite sides.

• Option 4: \(\angle S\cong\angle Q\) (opposite - angles) and \(\overline{TQ}\cong\overline{SR}\) (opposite - sides).

Answer:

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