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

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Congruent Chords Theorem" ], "new_concepts": [], "current_concepts": [ "Circle Geometry", "Congruent Chords Theorem", "Central Angle Theorem" ] } ## Identify congruent central angles $$ \angle PTQ \cong \angle STR $$ ## Relate central angles to intercepted arcs $$ m\widehat{PQ} = m\angle PTQ,\quad m\widehat{SR} = m\angle STR \implies \widehat{PQ} \cong \widehat{SR} $$ ## Apply the Congruent Chords Theorem $$ \widehat{PQ} \cong \widehat{SR} \implies \overline{QP} \cong \overline{SR} $$

(A) $\overline{QP}$ and $\overline{SR}$

(B) $\overline{QR}$ and $\overline{PQ}$

(C) $\overline{PR}$ and $\overline{RS}$

(D) $\overline{PR}$ and $\overline{PS}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Congruent Chords Theorem" ] }

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Question

Explanation:

Identify congruent central angles

Relate central angles to intercepted arcs

Apply the Congruent Chords Theorem

Answer:

Identify congruent central angles

Relate central angles to intercepted arcs

Apply the Congruent Chords Theorem