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?
Basic angle geometry question
3
$\begingroup$
geometry
2 Answers
5
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 $\alpha+\gamma=\beta$.
2
total degrees in a triangle = 180
$\beta_{supplement} = 180-\beta$
so
$180 = \gamma + \alpha + \beta_{supplement}$