Question

angles ptq and str are vertical angles and congruent. which chords are congruent? options: \\(\overline{qp}\\) and \\(\overline{sr}\\); \\(\overline{qr}\\) and \\(\overline{pq}\\); \\(\overline{pr}\\) and \\(\overline{rs}\\); \\(\overline{pr}\\) and \\(\overline{ps}\\)

Explanation:

In a circle, if two central angles (or vertical angles formed by intersecting chords) are congruent, then their intercepted chords are congruent. Angles \( \angle PTQ \) and \( \angle STR \) are vertical and congruent. The chords intercepted by these angles are \( \overline{QP} \) (intercepted by \( \angle PTQ \)) and \( \overline{SR} \) (intercepted by \( \angle STR \)). So by the theorem that congruent central (or vertical) angles intercept congruent chords, \( \overline{QP} \) and \( \overline{SR} \) are congruent.

Answer:

\( \boldsymbol{\overline{QP}} \) and \( \boldsymbol{\overline{SR}} \) (the first option: \( \overline{QP} \) and \( \overline{SR} \))