\section*{Exercise 3 - 290723  The image shows two points $P, Q$ and their image points $P^{\prime}, Q^{\prime}$ under a translation. A line $g$ is drawn through $Q^{\prime}$. Furthermore, let $\alpha$ and $\beta$ be the sizes of the angles marked in the image. Determine a size specification for $\gamma$, expressed in terms of $\alpha$ and $\beta$!}

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0516.jpg?height=339&width=434&top_left_y=293&top_left_x=1325)
If $S$ is the vertex of the marked angle of size $\beta$, then $\angle S P^{\prime} Q^{\prime}$ is the vertex angle of the marked angle of size $\alpha$. Therefore, $\angle S P^{\prime} Q^{\prime}=\alpha$.

According to the properties of any translation, the line through $P^{\prime}, Q^{\prime}$ is parallel to the line through $P, Q$. Therefore, by the corresponding angles theorem, $\angle P^{\prime} Q^{\prime} S=\gamma$. From the exterior angle theorem, applied to triangle $P^{\prime} Q^{\prime} S$, it follows that $\alpha+\gamma=\beta$ and thus $\gamma=\beta-\alpha$.