Question

question 10 of 10
what is the measure of \\( \angle xyz? \\)
a. \\( 75 ^ { \circ } \\)
b. \\( 108 ^ { \circ } \\)
c. \\( 54 ^ { \circ } \\)
d. \\( 33 ^ { \circ } \\)

Explanation:

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

The measure of an inscribed angle is half the measure of its intercepted arc. But here, since \( \angle XYZ\) and the angle formed by arcs \( \overset{\frown}{VX}\) and \( \overset{\frown}{WZ}\) (using the property that the measure of an angle formed by two intersecting chords \( \angle a=\frac{1}{2}(m\overset{\frown}{b} + m\overset{\frown}{c})\), where \(m\overset{\frown}{b}\) and \(m\overset{\frown}{c}\) are the measures of the intercepted arcs). However, if we use the property that the measure of an angle formed by two chords intersecting at the center (in this case, since \(Y\) is the center of the circle, \( \angle XYZ\) is related to the arcs. The sum of the arcs \( \overset{\frown}{VX}\) and \( \overset{\frown}{WZ}\) is \(33^{\circ}+75^{\circ}\). And the measure of \( \angle XYZ=\frac{1}{2}(33^{\circ}+75^{\circ})\) (by the angle - arc relationship for angles formed by two chords intersecting inside a circle: \( \angle=\frac{1}{2}(m\overset{\frown}{1}+m\overset{\frown}{2})\))

Step2: Calculate the value

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Answer:

C. \(54^{\circ}\)