Question

what is the measure of $\angle xyz$?\
\
a. $75^\circ$\
b. $108^\circ$\
c. $33^\circ$\
d. $54^\circ$

Explanation:

Step1: Recall inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. First, find the intercepted arc for $\angle XYZ$.

Step2: Calculate intercepted arc $XW$

The total circle is $360^\circ$. Subtract the given arcs: $\text{Arc } XW = 360^\circ - 75^\circ - 33^\circ - 180^\circ = 72^\circ$
(Note: $XZ$ and $VW$ are straight lines, so arc $VZ$ is $180^\circ$, leaving arc $XW$ as above)

Step3: Calculate $\angle XYZ$

$\angle XYZ$ intercepts arc $XW$ and arc $VZ$? No, correct: $\angle XYZ$ is formed by two chords, so its measure is half the sum of the intercepted arcs.
$\angle XYZ = \frac{1}{2} \times (\text{Arc } XZ' + \text{Arc } VW') = \frac{1}{2} \times (75^\circ + 33^\circ) = \frac{1}{2} \times 108^\circ = 54^\circ$

Answer:

D. $54^\circ$