5.3.5 Exercises

For example, the angles $\alpha$ and $\alpha \text{'}$ are corresponding angles. Likewise, angles $\beta$ and $\beta \text{'}$. For example, the angles $\alpha \text{'}$ and $\beta$ are alternate angles. Likewise, angles $\alpha$ and $\beta \text{'}$.

Draw a parallel to one of the sides of the triangle passing through the third vertex and consider the angles at this vertex.

Drawing a parallel to the side $c$ passing through the vertex $C$ one obtains an alternate angle $\alpha \text{'}$ to $\alpha$ and an alternate angle $\beta \text{'}$ to $\beta$. The angles $\alpha \text{'}$, $\gamma$, and $\beta \text{'}$ form a straight angle. Therefore, ${\displaystyle \alpha \text{'}+\gamma +\beta \text{'}=\pi \hspace{0.5em}.}$ Furthermore, it is known that $\alpha \text{'}=\alpha$ and $\beta \text{'}=\beta$. Hence, $\alpha +\gamma +\beta =\pi$.