Question

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

Explanation:

Step1: Recall circle chord theorem

In a circle, if two inscribed angles subtended by chords are congruent, and the central angles (here vertical angles $\angle PTQ \cong \angle STR$) imply the arcs subtended by the chords are congruent. Congruent arcs correspond to congruent chords.

Step2: Match angles to chords

$\angle PTQ$ subtends chord $\overline{QP}$, and $\angle STR$ subtends chord $\overline{SR}$. Since $\angle PTQ \cong \angle STR$, their subtended chords are congruent.

Answer:

$\overline{QP}$ and $\overline{SR}$