Question

segment $rs$ is congruent to segment $df$.
which congruence statement is true?
$\bigcirc widehat{rs} cong widehat{rd}$
$\bigcirc widehat{rs} cong widehat{sfd}$
$\bigcirc widehat{rs} cong widehat{sf}$
$\bigcirc widehat{rs} cong widehat{df}$

Explanation:

Step1: Recall circle chord theorem

In a circle, congruent chords correspond to congruent minor arcs.

Step2: Match given congruent chords

We are given $\overline{RS} \cong \overline{DF}$. By the theorem, their corresponding minor arcs $\widehat{RS}$ and $\widehat{DF}$ are congruent.

Step3: Eliminate incorrect options

• $\widehat{RS} \cong \widehat{RD}$: No evidence $\overline{RS} \cong \overline{RD}$, so false.
• $\widehat{RS} \cong \widehat{SFD}$: $\widehat{SFD}$ is a major arc, $\widehat{RS}$ is minor, so false.
• $\widehat{RS} \cong \widehat{SF}$: No evidence $\overline{RS} \cong \overline{SF}$, so false.

Answer:

$\boldsymbol{\widehat{RS} \cong \widehat{DF}}$