Question

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

Explanation:

Step1: Recall Vertical Angles and Chords

Vertical angles are congruent. In a circle, if two central angles (or angles formed by intersecting chords) are congruent, their intercepted chords are congruent. Angles \( \angle PTQ \) and \( \angle STR \) are vertical angles, so they are congruent. The chords intercepted by these angles: \( \angle PTQ \) intercepts \( \overline{PQ} \) (wait, no, \( \angle PTQ \) is formed by chords \( PQ \) and \( PR \)? Wait, no, looking at the diagram, \( T \) is the center? Wait, \( T \) is the intersection of chords \( PR \) and \( QS \). So \( \angle PTQ \) and \( \angle STR \) are vertical angles. The intercepted arcs: the angle \( \angle PTQ \) intercepts arc \( PQ \), and \( \angle STR \) intercepts arc \( SR \)? Wait, no, when two chords intersect at the center (if \( T \) is the center), then congruent central angles intercept congruent chords. Wait, maybe \( T \) is the center. So chords \( PQ \) and \( SR \): the angles \( \angle PTQ \) and \( \angle STR \) are congruent, so their intercepted chords \( QP \) (which is \( PQ \)) and \( SR \) should be congruent. Let's check the options. The second option is \( \overline{QP} \) and \( \overline{SR} \). Let's verify: if \( \angle PTQ \cong \angle STR \) (vertical angles), then the chords they subtend (intercept) are congruent. So chord \( QP \) and chord \( SR \) should be congruent.

Step2: Evaluate Each Option

• Option 1: \( \overline{PR} \) and \( \overline{RS} \): \( PR \) is a diameter (if \( T \) is center), \( RS \) is a chord, not necessarily congruent.
• Option 2: \( \overline{QP} \) and \( \overline{SR} \): As vertical angles \( \angle PTQ \cong \angle STR \), their intercepted chords \( QP \) and \( SR \) are congruent (since in a circle, congruent central angles intercept congruent chords).
• Option 3: \( \overline{QR} \) and \( \overline{PQ} \): No reason for these to be congruent.
• Option 4: \( \overline{PR} \) and \( \overline{PS} \): \( PR \) is a diameter, \( PS \) is a chord, not congruent.

Answer:

B. \( \overline{QP} \) and \( \overline{SR} \) (assuming the options are labeled as A, B, C, D with A being first, B second, etc. So the second option: \( \overline{QP} \) and \( \overline{SR} \))