Question

question 1 of 10
what is the measure of \\( \angle xyz \\)?
a. \\( 72 ^ { \circ } \\)
b. \\( 17 ^ { \circ } \\)
c. \\( 55 ^ { \circ } \\)
d. \\( 36 ^ { \circ } \\)

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. But here, we use the property that the measure of an angle formed by two chords intersecting at the center of a circle is related to the arcs. However, more accurately, if we consider the formula for the angle formed by two chords: \(\angle XYZ=\frac{1}{2}(\overset{\frown}{VW}+\overset{\frown}{XZ})\). Wait, no, actually, if we use the property that the measure of an inscribed angle \(\angle V\) intercepts arc \(WX\) and \(\angle XYZ\) is related. Wait, no, another approach: The sum of the measures of arcs in a circle is \(360^{\circ}\). But we can also use the property that \(\angle XYZ=\frac{1}{2}(\overset{\frown}{VW}-\overset{\frown}{XZ})\) is wrong. Wait, no, when two secants (or chords) form an angle outside the circle, \(\theta=\frac{1}{2}(\text{larger arc}-\text{smaller arc})\), but when the vertex is at the center, no. Wait, actually, we can use the fact that \(\angle XYZ\) is an inscribed - angle - related. Wait, no, if we consider the following: The measure of an inscribed angle \(\angle V\) intercepts arc \(WX\) and \(\angle XYZ\) is related. Wait, no, the correct formula is \(\angle XYZ=\frac{1}{2}(\overset{\frown}{VW}+\overset{\frown}{XZ})\) is incorrect. Wait, actually, we know that \(\angle XYZ=\frac{1}{2}(55^{\circ}+17^{\circ})\).

Step2: Calculate the measure of \(\angle XYZ\)

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Answer:

D. \(36^{\circ}\)