Question

quadrilateral rstq is a parallelogram. which of the following relationships must be true? \\(\overline{rs} \cong \overline{st}\\), \\(\angle r \cong \angle q\\), \\(\angle s \cong \angle q\\), \\(\overline{tq} \cong \overline{ts}\\) (with a diagram of parallelogram rstq)

Explanation:

In a parallelogram, opposite angles are congruent. Quadrilateral RSTQ is a parallelogram, so angle S and angle Q are opposite angles. Thus, \( \angle S \cong \angle Q \) must be true. Let's analyze other options:

• Option \( \overline{RS} \cong \overline{ST} \): In a general parallelogram, adjacent sides are not necessarily congruent (that's a rhombus property), so this is not always true.
• Option \( \angle R \cong \angle Q \): In a parallelogram, consecutive angles are supplementary, not necessarily congruent (unless it's a rectangle), so this is not always true.
• Option \( \overline{TQ} \cong \overline{TS} \): These are adjacent sides, and there's no property of parallelograms that makes adjacent sides congruent by default.

Answer:

\( \boldsymbol{\angle S \cong \angle Q} \) (the option with \( \angle S \cong \angle Q \))