Question

what is the measure of $\angle xyz$?
a. $64^\circ$
b. $108^\circ$
c. $116^\circ$
d. $72^\circ$

Explanation:

Step1: Find arc VW measure

The total circle is $360^\circ$. Subtract given arcs:
$360^\circ - 64^\circ - 152^\circ = 144^\circ$

Step2: Identify inscribed angle rule

$\angle XYZ$ is an inscribed angle intercepting arc VWX. Arc VWX = arc VW + arc XV:
$144^\circ + 64^\circ = 208^\circ$
Wait, correct rule: inscribed angle over arc outside the angle uses $\frac{1}{2}$(difference of arcs). The angle intercepts arc XZ ($64^\circ$) and arc VWZ ($360^\circ-64^\circ=296^\circ$). The formula for angle formed by two chords inside a circle is $\frac{1}{2}$(sum of intercepted arcs). $\angle XYZ$ intercepts arc XZ and arc VW:
$\frac{1}{2}(64^\circ + 144^\circ) = \frac{1}{2}(208^\circ) = 104^\circ$
Wait, correct approach: Arc XW is a diameter? No, arc WZ is $152^\circ$, so arc VW is $360-64-152=144$. $\angle XYZ$ is formed by chords YZ and YX. The intercepted arcs are XZ ($64^\circ$) and VW ($144^\circ$). The measure of an angle formed by two intersecting chords is half the sum of the intercepted arcs:
$\frac{1}{2}(64^\circ + 144^\circ) = 104^\circ$
Wait, no, alternative: Arc XZ is $64^\circ$, so the central angle is $64^\circ$. $\angle XYZ$ is an angle at the intersection of chords VZ and XW. The formula is $\frac{1}{2}$(arc XZ + arc VW). Arc VW = $360-64-152=144$.
$\frac{1}{2}(64 + 144) = 104$
Wait, but options have 108. Wait, miscalculation: $360-64-152=144$ is wrong. $64+152=216$, $360-216=144$. Correct. Wait, maybe the angle is formed by tangent? No, VZ is a chord. Wait, no: $\angle XYZ$ intercepts arc XWZ? No, XW is a line through Y, so Y is the center? Oh! Y is the center. Then $\angle XYZ$ is a central angle intercepting arc XZ? No, arc XZ is $64^\circ$, but that's not matching. Wait, no, if Y is the center, then arc XZ is $64^\circ$, so $\angle XYZ$ would be $64^\circ$, but that's option A. No, wait, the arc XZ is labeled $64^\circ$, so the central angle is $64^\circ$, but $\angle XYZ$ is not that. Wait, no, VZ is a chord, XW is a diameter. Arc WZ is $152^\circ$, so arc XZ is $180-152=28^\circ$? No, the label $64^\circ$ is arc XV. Oh! I misread: arc XV is $64^\circ$, arc WZ is $152^\circ$. Then arc XZ is unknown, arc VW is unknown. XW is a diameter, so arc XW is $180^\circ$. Arc XW = arc XV + arc VW, so arc VW = $180-64=116^\circ$. Arc VZ = arc VW + arc WZ = $116+152=268^\circ$, no. $\angle XYZ$ is an inscribed angle intercepting arc XZ. Arc XZ = $360-64-116-152=28^\circ$? No, XW is a diameter, so arc XW is $180$, so arc XZ = $180-152=28^\circ$. Then $\angle XYZ$ is an angle formed by chord YZ and YX, with Y on XW. $\angle XYZ$ is equal to $\frac{1}{2}$(arc XZ + arc VW). Arc VW is $116^\circ$, arc XZ is $28^\circ$. $\frac{1}{2}(116+28)=72^\circ$? No, option D is 72. Wait, no, if Y is the center, then $\angle XYZ$ is central angle for arc XZ. Arc XZ = $180-152=28$, no. Wait, correct formula: When two chords intersect at Y, $\angle XYZ = \frac{1}{2}$(arc XZ + arc VW). Arc VW = $360-64-152-arc XZ$. Since XW is a straight line, arc XW is $180=64+arc VW$, so arc VW=116. Arc XZ=180-152=28. $\frac{1}{2}(28+116)=72^\circ$. Yes, that's option D.
Wait, final correct steps:

Step1: Find arc VW measure

XW is a straight line (diameter), so arc XW = $180^\circ$.
$180^\circ - 64^\circ = 116^\circ$

Step2: Find arc XZ measure

Arc WZ is $152^\circ$, so arc XZ = $180^\circ - 152^\circ = 28^\circ$

Step3: Calculate $\angle XYZ$

Angle from intersecting chords: $\frac{1}{2}$(sum of intercepted arcs)
$\frac{1}{2}(116^\circ + 28^\circ) = \frac{1}{2}(144^\circ) = 72^\circ$

Answer:

D. $72^\circ$