Yes, what you have is correct, the angle is not determined another way.

However, when talking about the angle of a ray or a line coming out of the origin, there are multiple (infinitely many) different angles that will correspond with the direction of the line.  Since a circle has 360 degrees, if you spin around the origin 360 degrees clockwise or counter clockwise (cw or ccw) you'll be right back where you started, since you simply went a full circle.  Of course, you could spin around multiple full circles (in either direction, cw or ccw) and all will still lead you back to the "terminal side" of the original angle where you started.

So, for example, your y=x in the first quadrant corresponds (ie, is described by) to angles of any of the following degrees:

45, 405, 765, 1125, ......


as well as

-315, -675, -1035, -1395, ......


because they all differ by a multiple of 360 (a multiple of full circles).  All of these angles are therefore called "coterminal", having the same terminal side.  The positive and negative parts of the angles simply denote the direction of the angle as either ccw or cw, respectively.

Well, you've seem to have made a 'simple problem' into a complex one. This info you've posted may come in handy (to me) in the future. But for now, I think my reasoning above (and below) is a logical way of stating the angle of a Line that passes through origin, doesn't have 1 angle.

When I basically drew a 'straight line'  through Origin that bisected the 1st quadrant into 2 equal parts. This indicated (to me) that the line across the 1st quadrant must be a 45 deg. angle, relative to both the X & Y axis.

Then, if I place a compass at origin, and its 'Pencil end' along the positive X axis. Then if I turn it left, until I reaches the lower end of the 'line' I've described......I will have arrived at 225 deg. (180 deg. + 45 deg. .....which equals an angle of 225 deg.  Meaning the upper half of the line is at 45 deg.  and the lower half is at 225 deg.  ...as relative to the
positive side of X axis.