In a circle, the inscribed angle theorem states that the measure of an inscribed angle is half the measure of the arc that it intercepts.
Here, triangle XYZ is inscribed in a circle with diameter XZ, and the angle measure at point X is given as 59°.
Since XZ is the diameter, the inscribed angle at Y (which intercepts arc XY) will be equal to half the measure of arc XZ. According to the properties of inscribed angles, angle Y is also a right angle (90°) because it subtends a semicircle.
Therefore, since angle Y is equal to half the measure of arc XZ, we can set up the following equation:
\[ \text{Angle Y} = \frac{1}{2} \text{arc XZ} \] \[ 90^\circ = \frac{1}{2} \text{arc XZ} \]
This means that: \[ \text{arc XZ} = 90^\circ \times 2 = 180^\circ \]
Now, the total circumference of the circle is 360°, which means the remainder of the circle (the entire arc around the circle excluding XZ) will add up to 360° as well.
To find the measure of arc XY, we know that: \[ \text{arc XZ} = \text{arc XY} + \text{arc YZ} \]
Since S is the opposite angle: \[ \text{arc YZ} = \text{arc XZ} - \text{arc XY} \]
To find the measure of arc XY specifically, we note that angle X (which measures 59°) also subtends arc XY. Thus:
\[ \text{angle X} = \frac{1}{2} \text{arc XY} \] \[ 59^\circ = \frac{1}{2} \text{arc XY} \] \[ \text{arc XY} = 59^\circ \times 2 = 118^\circ \]
Thus, the measure of arc XY is \(\boxed{118^\circ}\).