The story begins like..
I was reading through some pages of my highschool textbook....when I found this 
The question is straightforward: What I have studied till today tells me that, for any angle of inclination $\theta$, we must have $\theta\in[0^\circ,180^\circ]$. However my textbook compels me to negative angle of inclination. I do feel that anyone of you might say that it is the exact same thing, but it is a highschool student this side and thus a sincere explanation would be well appreciated.
That was the first question if they are meant to be same thing or not? The second that follows up is: If yes, then how are they same? If No, then why is it even printed in our textbooks.?
Furthermore, I must say that, in my opinions, it is really bad of writers that why should they go for such an out of the box, moreover an illogical set of knowledge that makes no sense due to the definition of $\theta$ in $[0^\circ,180^\circ]$. What do you guys say?
I'm not sure who told you that inclination is always in the range $[0^\circ,180^\circ]$, but you are now encountering a pattern that you will encounter often if you continue to study mathematics, which is that definitions can be extended.
You may also find that at some point, definitions may be "forgotten." I have seen mathematical writing in which there is an "angle $\theta$ counterclockwise from the positive $x$ axis" where you might have written more compactly "inclination $\theta$." The writer uses more verbose language because this mathematics is written at a level where it is not considered worthwhile to remember every definition that a reader might have encountered in a high school textbook and because a writer cannot rely on all of those textbooks to have made the exact same definition.
By the way, when you see an "angle $\theta$ counterclockwise from the positive $x$ axis" in higher mathematics, it is understood that a negative value of $\theta$ means the angle is measured clockwise, and that an angle greater than $180^\circ$ is possible.
The negative angles are very convenient when we want to use the arc tangent function to convert the slope of a line to an angle, because the arc tangent function gives a negative number as output when the input number is negative.
The question in my mind is really why someone would tell you that an inclination angle must always be in the range $[0^\circ,180^\circ]$. I have seen it argued that if students are shown a less restrictive definition of a concept at first, they will be confused by it and will not learn the material. Therefore a textbook writer will sometimes artificially restrict a definition due to the belief that this makes the material easier to learn. The consequence is that the student is confused later when they encounter the same word or concept in a less restrictive context.
In summary, I think the reason this book contradicts your earlier book is because the author of the earlier book did not think you were ready to understand a negative inclination, whereas the author of this book thinks you are ready.