4. In the diagram, $A B C D$ is a rectangle, $P$ is on $B C, Q$ is on $C D$, and $R$ is inside $A B C D$. Also, $\angle P R Q=30^{\circ}$, $\angle R Q D=w^{\circ}, \angle P Q C=x^{\circ}, \angle C P Q=y^{\circ}$, and $\angle B P R=z^{\circ}$. What is the value of $w+x+y+z$ ?
4. Since $\angle D Q C$ is a straight angle, $\angle R Q P=180^{\circ}-\angle R Q D-\angle P Q C=180^{\circ}-w^{\circ}-x^{\circ}$.
Since $\angle B P C$ is a straight angle, $\angle R P Q=180^{\circ}-\angle R P B-\angle Q P C=180^{\circ}-z^{\circ}-y^{\circ}$.
The sum of the measures of the angles in $\triangle P Q R$ is $180^{\circ}$, and so
$$
\begin{aligned}
\angle R Q P+\angle R P Q+\angle P R Q & =180^{\circ} \\
\left(180^{\circ}-w^{\circ}-x^{\circ}\right)+\left(180^{\circ}-z^{\circ}-y^{\circ}\right)+30^{\circ} & =180^{\circ} \\
390^{\circ}-w^{\circ}-x^{\circ}-y^{\circ}-z^{\circ} & =180^{\circ} \\
210^{\circ} & =w^{\circ}+x^{\circ}+y^{\circ}+z^{\circ}
\end{aligned}
$$
and so $w+x+y+z=210$.
ANSWER: 210