In the book of Linear Algebra by Werner Greub, at page 202 Q.11.a, it is asked that
Let $x,y,z$ be three vectors of a plane such that $x$ and $y$ are linearly interdependent and that $x+y+z = 0$.
a-) Prove that the ordered pairs $x,y; y,z; z,x$ represent the same orientation. Then show that $$\theta(x,y) + \theta(y,z) + \theta(z,x) = 2\pi$$ where the angles refer to the above orientation.
I can see that they represent the same orientation but couldn't show the followed result. I was hoping you can help me on that point.