Given:
A circle with center Z and the points W, X, Y are on the circumference of the circle.
To find:
The measure of angle WXY.
Solution:
From the given figure, we get
![Arc(XY)=156^\circ](https://tex.z-dn.net/?f=Arc%28XY%29%3D156%5E%5Ccirc)
![Arc(WX)=86^\circ](https://tex.z-dn.net/?f=Arc%28WX%29%3D86%5E%5Ccirc)
The measure of arc of the complete circle is 360 degrees.
![Arc(WX)+Arc(XY)+Arc(WY)=360^\circ](https://tex.z-dn.net/?f=Arc%28WX%29%2BArc%28XY%29%2BArc%28WY%29%3D360%5E%5Ccirc)
![86^\circ+156^\circ+Arc(WY)=360^\circ](https://tex.z-dn.net/?f=86%5E%5Ccirc%2B156%5E%5Ccirc%2BArc%28WY%29%3D360%5E%5Ccirc)
![Arc(WY)=360^\circ-86^\circ-156^\circ](https://tex.z-dn.net/?f=Arc%28WY%29%3D360%5E%5Ccirc-86%5E%5Ccirc-156%5E%5Ccirc)
![Arc(WY)=118^\circ](https://tex.z-dn.net/?f=Arc%28WY%29%3D118%5E%5Ccirc)
The measure of arc WY is 118 degrees. It means the central angle on this arc is also 118 degrees.
![m\angle WZY=118^\circ](https://tex.z-dn.net/?f=m%5Cangle%20WZY%3D118%5E%5Ccirc)
According to the central angle theorem, the inscribed angle on an arc is always half of its central angle.
![m\angle WXY=\dfrac{m\angle WZY}{2}](https://tex.z-dn.net/?f=m%5Cangle%20WXY%3D%5Cdfrac%7Bm%5Cangle%20WZY%7D%7B2%7D)
![m\angle WXY=\dfrac{118^\circ}{2}](https://tex.z-dn.net/?f=m%5Cangle%20WXY%3D%5Cdfrac%7B118%5E%5Ccirc%7D%7B2%7D)
![m\angle WXY=59^\circ](https://tex.z-dn.net/?f=m%5Cangle%20WXY%3D59%5E%5Ccirc)
Therefore, the correct option is A.