Question

what is the measure of \\( \overleftrightarrow{xy} \\)?

Explanation:

Step1: Use the property of vertical angles and arc - angle relationship

In a circle, if two chords intersect at the center \(Z\), then vertical angles are equal. Also, the measure of an arc is related to the central angle. The sum of central angles around a point is \(360^{\circ}\). But here, we can use the fact that \(\angle XZY=\angle UZV = 42^{\circ}\) (vertical angles). Let the measure of arc \(\overset{\frown}{XY}\) be \(m\overset{\frown}{XY}\), the measure of arc \(\overset{\frown}{UV}\) be \(m\overset{\frown}{UV}=38^{\circ}\).

We know that \(m\overset{\frown}{XY}+m\overset{\frown}{UV}+ 2\times42^{\circ}=360^{\circ}- 2\times(180^{\circ})\) (not needed). Another approach: Since the sum of central angles in a circle: \(m\overset{\frown}{XY}+m\overset{\frown}{UV}+2\times42^{\circ}=360^{\circ}- 2\times(180^{\circ})\) (simpler way). But more directly, using the property that \(m\overset{\frown}{XY}=180^{\circ}-42^{\circ}-92^{\circ}\) (no). Wait, correct formula: The sum of central angles around \(Z\): \(m\overset{\frown}{XY}+m\overset{\frown}{UV}+2\times42^{\circ}=360^{\circ}- 2\times(180^{\circ})\) (wrong). Correct: In a circle, the sum of central angles: Let \(m\overset{\frown}{XY}=x\). We know that \(x + 38^{\circ}+42^{\circ}+42^{\circ}=180^{\circ}\) (because the two intersecting chords form two pairs of vertical angles and the sum of angles around a point on a straight - line (diameter - like concept here) is \(180^{\circ}\)).

Step2: Solve for \(x\)

$$
LATEXBLOCK0
$$

Answer:

B. \(46^{\circ}\)