1. The problem describes a coordinate system with an angle $\theta$ formed at the origin between the $x$-axis and a ray extending upward right. 2. Points $N$, $M$, and $M'$ lie on vertical line segments above the $x$-axis. 3. Point $p$ and angle $\beta$ are also mentioned, with a small triangle at $p$ containing $\beta$ adjacent to the arc through $p$ and $N$. 4. To understand how to find or interpret $\theta$, note it measures the angle between the positive $x$-axis and the line from the origin to the point on the ray. 5. If the ray's endpoint coordinates are $(x, y)$, then $$\theta = \arctan\left(\frac{y}{x}\right).$$ 6. This formula gives the angle $\theta$ in radians between the $x$-axis and the line segment from the origin to $(x,y)$. 7. The angle $\beta$ formed at $p$ relates to other geometric properties, but for $\theta$ the key is its definition as the angle from the positive $x$-axis to the ray. Final answer: The angle $\theta$ is given by $$\theta = \arctan\left(\frac{y}{x}\right).$$ This means that to find $\theta$ you need the coordinates of the point on the line, and then compute the inverse tangent of $y$ over $x$.