Question

question 7 of 10
what is the measure of \\( \angle x y z \\)?
a. \\( 72 ^ { \circ } \\)
b. \\( 108 ^ { \circ } \\)
c. \\( 64 ^ { \circ } \\)
d. \\( 116 ^ { \circ } \\)

Explanation:

Step1: Calculate the total degrees of a circle

The total degrees in a circle is \(360^{\circ}\).

Step2: Set up the equation

Let \(\angle XYZ = x\). We know that \(64^{\circ}+152^{\circ}+x + \angle VYW=360^{\circ}\). Since \(\angle VYW\) and \(\angle XYZ\) are vertical angles (not necessary here as we can also use the fact that the sum of arcs and central - angle relationships. The sum of the given arcs \(64^{\circ}+152^{\circ}=216^{\circ}\). The remaining two central angles (which are equal as \(VY = ZY\) and \(XY = WY\) in the circle) sum to \(360 - 216=144^{\circ}\). But a simpler way: The sum of the arcs not adjacent to \(\angle XYZ\) is \(64^{\circ}\). Using the property that the sum of arcs in a circle is \(360^{\circ}\) and for central angles. Another approach: The sum of arcs \(64^{\circ}+152^{\circ}\) and the two arcs corresponding to \(\angle XYZ\) (let's call the arc opposite to \(64^{\circ}\) as \(a\) and opposite to \(152^{\circ}\) as \(b\)). But since \(VY = ZY\) and \(XY = WY\) (radii of the circle), we can also use the formula for the sum of central angles. The sum of the arcs \(64^{\circ}+152^{\circ}\) and the two arcs formed by \(\angle XYZ\) (let the measure of \(\angle XYZ\) be \(x\)). The sum of arcs in a circle is \(360^{\circ}\). We know that the sum of the non - \(\angle XYZ\) related arcs \(64^{\circ}\). Wait, a better formula: The sum of central angles in a circle is \(360^{\circ}\). Let \(\angle XYZ=\theta\). The sum of the arcs: The arc opposite to \(\theta\) (if we consider the circle) but actually, since \(VY = ZY\) and \(XY = WY\) (radii), we can use the fact that \(\angle XYZ=\frac{1}{2}(360-(64 + 152))\) is wrong. Wait, no. The correct formula for central angles: The sum of central angles in a circle is \(360^{\circ}\). We know that \(\angle XYZ\) and its adjacent angles. Since \(VY = ZY\) and \(XY = WY\) (radii of the circle), we can also use the property that the sum of angles around a point \(Y\) is \(360^{\circ}\). But if we assume that the two non - given arcs are equal (because of the symmetry of the circle with radii). The sum of the given arcs \(64^{\circ}+152^{\circ}=216^{\circ}\). The remaining \(360 - 216 = 144^{\circ}\) is split between two equal angles (because \(VY = ZY\) and \(XY = WY\) (radii)). But another way: The measure of \(\angle XYZ\) can be found by \(180-\frac{64}{2}\) (wrong). Wait, correct formula: The sum of arcs in a circle is \(360^{\circ}\). Let’s use the property that the sum of central angles. We know that \(\angle XYZ\) and its supplementary - like angles. Wait, no. The correct approach: The sum of arcs \(64^{\circ}+152^{\circ}\) and the two arcs formed by \(\angle XYZ\) (since \(VY = ZY\) and \(XY = WY\) (radii)). The sum of arcs \(=360^{\circ}\). Let \(m\angle XYZ=\alpha\). The arc corresponding to \(\alpha\) (if we consider the circle) but actually, since \(VY = ZY\) and \(XY = WY\) (radii), we can use the fact that \(\angle XYZ = 180-\frac{64}{2}\) (wrong). Wait, correct: The sum of the arcs \(64^{\circ}+152^{\circ}\) and the two arcs (let’s call them \(a\) and \(a\)) (because of the symmetry of the circle with radii). \(64 + 152+2a=360\), \(2a=360-(64 + 152)=144\), \(a = 72\). But \(\angle XYZ=180 - 72=108^{\circ}\)

Answer:

B. \(108^{\circ}\)