Question

what is the measure of
a. 128°
b. 38°
c. 142°
d. 52°

Explanation:

Step1: Find inscribed angle at X

The angle between tangent \(XY\) and chord \(XW\) equals the inscribed angle subtended by \(XW\). So the inscribed angle \(\angle XVW = 38^\circ\).

Step2: Find central angle for arc \(XW\)

Central angle is twice the inscribed angle.
\(\angle XOW = 2 \times 38^\circ = 76^\circ\)

Step3: Calculate arc \(YZ\)

We know \(\angle VOZ = 90^\circ\) and \(\angle VOX = 90^\circ\). The total around center \(O\) is \(360^\circ\).
Arc \(YZ = 360^\circ - 90^\circ - 90^\circ - 76^\circ = 104^\circ\)

Step4: Find inscribed angle \(\angle YXZ\)

Inscribed angle is half its subtended arc.
\(\angle YXZ = \frac{1}{2} \times 104^\circ = 52^\circ\)

Step5: Calculate \(\angle XZ\) (angle \(\angle YZX\))

In \(\triangle XYZ\), \(\angle XYZ = 90^\circ\) (angle subtended by diameter \(XZ\)).
\(\angle YZX = 180^\circ - 90^\circ - 52^\circ = 38^\circ\)
Wait, correction: The tangent-chord theorem gives \(\angle WXY = 38^\circ\), and \(\angle XYW = 90^\circ\) (since \(YW\) is perpendicular to diameter \(XZ\)). In \(\triangle XYW\), \(\angle YXW = 90^\circ - 38^\circ = 52^\circ\). The angle \(\angle YZX\) is equal to \(\angle YXW\) as they subtend arc \(YW\). No, correct approach: The angle between tangent \(XY\) and secant \(XZ\) is half the difference of the intercepted arcs.
\(\angle YXZ = \frac{1}{2} (\text{arc } YZ - \text{arc } WX)\)
Arc \(WX = 76^\circ\), arc \(YZ = 360^\circ - 90^\circ - 90^\circ -76^\circ=104^\circ\)
\(\angle YXZ = \frac{1}{2}(104^\circ -76^\circ)=14^\circ\) no, wrong. Correct: The tangent angle \(\angle WXY = 38^\circ = \frac{1}{2} \text{arc } WY\), so arc \(WY=76^\circ\). Arc \(WZ = 90^\circ +90^\circ=180^\circ\) (semicircle). Arc \(YZ=180^\circ -76^\circ=104^\circ\). The inscribed angle \(\angle YXZ = \frac{1}{2} \text{arc } YW = 38^\circ\)? No, the question asks for \(\angle XZ\), which is \(\angle YZX\). \(\angle YZX = \frac{1}{2} \text{arc } YW = 38^\circ\)

Answer:

B. \(38^\circ\)