Basic angle geometry question 
I've faced a question that needed to find angle $\gamma$ as a part of it and the solution came from $\gamma= \beta - \alpha$. How did the book arrive to such conclusion, and does this apply for every similar case?

The angle "next to" $\beta$ (supplementary to $\beta$) is $180^\circ-\beta$.  The sum of the angles of a triangle is $180^\circ$, so
$$\alpha+(180^\circ-\beta)+\gamma=180^\circ.$$
Remark: This is an often-used result. Usually it is stated as follows. The external angle ($\beta$) at a vertex is the sum of the internal angles at the other two vertices. So $\alpha+\gamma=\beta$.
