Question

which of the following arcs are congruent in the circle below? image of circle with points w, x, o, y, z; ∠wox and ∠yoz are 27° options: a. $widehat{wx} cong widehat{xz}$; b. $widehat{wx} cong widehat{yz}$; c. $widehat{wy} cong widehat{xz}$; d. $widehat{wxy} cong widehat{yw}$

Explanation:

Step1: Recall Arc Congruence Rule

In a circle, arcs are congruent if their central angles are equal (since all radii are equal, equal central angles imply equal arc lengths).

Step2: Analyze Central Angles

• For arc \(\widehat{WX}\), the central angle is \(\angle WOX = 27^\circ\).
• For arc \(\widehat{YZ}\), the central angle is \(\angle YOZ = 27^\circ\).
• Since \(\angle WOX=\angle YOZ = 27^\circ\), their corresponding arcs \(\widehat{WX}\) and \(\widehat{YZ}\) are congruent.

• Check other options:
• Option A: \(\widehat{WX}\) and \(\widehat{XZ}\) have different central angles (not \(27^\circ\) for \(\widehat{XZ}\)).
• Option C: \(\widehat{WY}\) and \(\widehat{XZ}\) have different central angles.
• Option D: \(\widehat{WXY}\) is a major arc and \(\widehat{YW}\) (assuming typo, but conceptually different) and their central angles won't be equal.

Answer:

B. \(\widehat{WX} \cong \widehat{YZ}\)