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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The angle $y$ and the angle which is $31 ^\circ$ are vertical angles. According to the Vertical Angles Theorem the angles are congruent and have the same measure.

The sum of the green angles is $31 ^\circ+31^\circ=62^\circ.$ Since a complete rotation is $360 ^\circ,$ the remaining $298 ^\circ$ are distributed over the other two unknown angles. These also form a pair of vertical angles and are congruent. Thus, their measure is half of $298^\circ.$ $\dfrac{298^\circ}{2\,\,}=149^\circ$ Thus, both $\angle x$ and $\angle z$ have the measure $149^\circ.$

As in the other solution, we can determine the $m\angle y$ to be $31^\circ.$

The $\angle x$ and any of the angles with the measure $31^\circ$ are supplementary angles. Thus, the sum of their measures is $180^\circ.$ $x+31^\circ=180^\circ \quad \Leftrightarrow \quad x=149^\circ.$ $x$ and $z$ are vertical angles and are congruent. Therefore, $m\angle z=149^\circ.$