Question

angles ptq and str are vertical angles and congruent.
which arcs are congruent?
$overarc{pq}$ and $overarc{sr}$
$overarc{sp}$ and $overarc{pr}$
$overarc{sp}$ and $overarc{sr}$
$overarc{pq}$ and $overarc{qr}$

Explanation:

Step1: Recall Vertical Angles and Arcs

Vertical angles (∠PTQ and ∠STR) are congruent. In a circle, congruent central angles subtend congruent arcs. ∠PTQ subtends arc \( \overarc{PQ} \), and ∠STR subtends arc \( \overarc{SR} \).

Step2: Analyze Each Option

• Option 1: \( \overarc{PQ} \) and \( \overarc{SR} \) – Since ∠PTQ ≅ ∠STR (vertical angles), their subtended arcs should be congruent.
• Option 2: \( \overarc{SP} \) and \( \overarc{PR} \) – No info shows their subtending angles are congruent.
• Option 3: \( \overarc{SP} \) and \( \overarc{SR} \) – Subtending angles not shown congruent.
• Option 4: \( \overarc{PQ} \) and \( \overarc{QR} \) – Subtending angles not congruent.

Answer:

\( \overarc{PQ} \) and \( \overarc{SR} \) (the first option)