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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?

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up vote 5 down vote accepted

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 $\alpha+\gamma=\beta$.

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total degrees in a triangle = 180

$$\beta_{supplement} = 180-\beta$$


$$180 = \gamma + \alpha + \beta_{supplement}$$

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