Geometry problem with circles and arcs

Question

In the diagram, $\angle U = 30^\circ$, arc $XY$ is $170^\circ$, and arc $VW$ is $110^\circ$. Find arc $WY$, in degrees.

I got $55^\circ$ but apparently it's wrong.

Answer 1

Draw the figure such that $XY$ is horizontal. Then draw provisional $V'W'$ forming an angle of $110^\circ$ at the center $O$ of the circle, so that $V'W'$ is horizontal as well. Now rotate $V'W'$ clockwise into the definitive position $VW$ so that at $U$ you obtain the angle $30^\circ$. Looking at all angles and half-angles in your figure then shows you that the angle you desire is $70^\circ$.

Answer 2

Construct $YV$ as the diameter of the circle. Then

\begin{align} \angle XOY&=170^\circ ,\\ \angle XWY&=85^\circ ,\\ \angle XYO=\angle OYX=\angle VWX &=5^\circ ,\\ \angle VOX&=10^\circ ,\\ \angle YOV&=70^\circ ,\\ \angle VYU&=60^\circ ,\\ \angle UWY&=90^\circ ,\\ \angle YUW&=30^\circ . \end{align}

Edit

Alternatively,

\begin{align} \triangle UWY:\quad \angle YUW&+\angle UWY+\angle WYU =180^\circ ,\\ \angle UWY&= \angle UWO+\angle OWY =125^\circ-\tfrac12\phi ,\\ \angle WYU&= \angle WUO+\angle OYU =95^\circ-\tfrac12\phi ,\\ 30^\circ+ 125^\circ&-\tfrac12\phi +95^\circ-\tfrac12\phi =180^\circ ,\\ \phi&=70^\circ . \end{align}

Answer 3

Use this fact about two secants that meet at a point outside the circle: $\newcommand{arc}{\mathop{\mathrm{arc}}}$ $$ \frac12(\arc WY - \arc XV) = \angle U = 30^\circ. $$

Also use the fact that the circle is divided into four arcs:

$$ \arc WY + \arc XV = 360^\circ - \arc XY - \arc VW = 360^\circ - 170 ^\circ - 110^\circ = 80^\circ. $$

Now you have two equations in just two unknowns, $\arc WY$ and $\arc XV.$ Solve for $\arc WY.$