As James K notes, when you are dealing with lines like this where the angle is an important defining feature it is simplest to use polar notation and refer to it as "A line segment with angle theta and radius r" (leaving off the part about "radius r" if the line extends forever).
If you must use the Cartesian coordinate system things are ...
That is a half-line from the point (x₀, y₀), making an angle of θ with the x-axis.
But if you are doing maths, it is easier to do maths and describe the line using polar coordinates or as a set of numbers on the Argand diagram.
The locus of points satisfying arg( z - x₀-y₀i ) = θ