Question

a circular token in a board game has two triangles drawn on it, as shown. for which of the following conditions would
\\( \widehat { a b } \cong \widehat { b c } \\) in circle \\( m \\)?
\\( \textcircled { a } \overline { c m } \cong \overline { a m } \\)
\\( \textcircled { b } \overline { a b } \cong \overline { b c } \\)
\\( \textcircled { c } \overline { m b } \cong \overline { m c } \\)
\\( \textcircled { d } \overline { m a } \cong \overline { m b } \\)

Explanation:

Step1: Recall the property of arcs and chords in a circle

In a circle, if two chords are congruent, then their corresponding arcs are congruent.

Step2: Analyze each option

• Option A: $\overline{CM}\cong\overline{AM}$ just tells that two radii are congruent, which doesn't directly relate to the congruence of arcs $\overset{\frown}{AB}$ and $\overset{\frown}{BC}$.
• Option B: If $\overline{AB}\cong\overline{BC}$, by the property that congruent chords correspond to congruent arcs in a circle, we can conclude that $\overset{\frown}{AB}\cong\overset{\frown}{BC}$.
• Option C: $\overline{MB}\cong\overline{MC}$ only shows two radii are congruent, not relevant to the arcs $\overset{\frown}{AB}$ and $\overset{\frown}{BC}$.
• Option D: $\overline{MA}\cong\overline{MB}$ only shows two radii are congruent, not relevant to the arcs $\overset{\frown}{AB}$ and $\overset{\frown}{BC}$.

Answer:

B. $\overline{AB}\cong\overline{BC}$