Calculate and theta, and then find the size of PQR to the nearest degree.

In $\triangle PQM$, the side opposite to $\theta$ is $PM = 9$, and the side adjacent to $\theta$ is $QM = 10$.
We use the tangent function:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{PM}{QM}$$ $$\tan(\theta) = \frac{9}{10}$$ $$\theta = \arctan\left(\frac{9}{10}\right)$$ $$\theta \approx 41.987^\circ$$

In $\triangle RQM$, the side opposite to $\phi$ is $MR = 23$, and the side adjacent to $\phi$ is $QM = 10$.
We use the tangent function:

$$\tan(\phi) = \frac{\text{opposite}}{\text{adjacent}} = \frac{MR}{QM}$$ $$\tan(\phi) = \frac{23}{10}$$ $$\phi = \arctan\left(\frac{23}{10}\right)$$ $$\phi \approx 66.501^\circ$$

The angle $\angle PQR$ is the sum of $\theta$ and $\phi$:

$$\angle PQR = \theta + \phi$$ $$\angle PQR \approx 41.987^\circ + 66.501^\circ$$ $$\angle PQR \approx 108.488^\circ$$

$$\angle PQR \approx 108^\circ$$

The values are:
$\boxed{\theta \approx 42.0^\circ}$ (to one decimal place)
$\boxed{\phi \approx 66.5^\circ}$ (to one decimal place)
The size of $\angle PQR$ to the nearest degree is $\boxed{108^\circ}$.