Question

identifying a true statement
in circle q, $angle rqs cong angle sqt$.
which statement must be true?
$overline{rt} cong overline{ut}$
$angle rqt cong angle rst$
$overline{rq} perp overline{qt}$
$overline{rs} cong overline{st}$

Explanation:

Step1: Recall circle chord-angle theorem

In a circle, if two central angles are congruent, their corresponding chords are congruent.

Step2: Match given condition to theorem

We know $\angle RQS \cong \angle SQT$, these are central angles corresponding to chords $\overline{RS}$ and $\overline{ST}$ respectively.

Step3: Evaluate other options

• $\widehat{RT} \cong \widehat{UT}$: No given info about $\angle RQT$ and $\angle UQT$, so unproven.
• $\angle RQT \cong \angle RST$: $\angle RST$ is an inscribed angle, $\angle RQT$ is central; they are only congruent if $\angle RQT=120^\circ$, which is not given.
• $\overline{RQ} \perp \overline{QT}$: No info that $\angle RQT=90^\circ$, so unproven.

Answer:

$\overline{RS} \cong \overline{ST}$