## Task 5 - 010525

Draw an arbitrary triangle and name its angles $\alpha, \beta$, and $\gamma$!

Construct with compass and straightedge outside the triangle the angle $\alpha+\beta+\gamma$! How large is the constructed angle likely to be?
The left image shows triangle $ABC$ with interior angles $\alpha, \beta$, and $\gamma$. To construct the sum of the angles $\alpha+\beta+\gamma$, one chooses an arbitrary point $P$ and from this point, a ray $s$. (right image)

To mark the points $A, B$, and $C$ of the triangle and the point $P$, draw circular arcs with the same radius. The intersection points of the circular arcs with the sides of the triangle are $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}$, and $C_{2}$. The intersection point of the circular arc with the ray $s$ is $D$.

The compass spans $A_{1} A_{2}, B_{1} B_{2}$, and $C_{1} C_{2}$ are transferred one after the other, starting from $D$, onto the circular arc around $P$ and obtain the intersection points $E, F$, and $G$. If $G$ is connected to $P$, then the angle $\angle D P G$ is the sum of the interior angles of the triangle. One can see that $D, P$, and $G$ lie on a straight line. The angle $\alpha+\beta+\gamma$ is thus a straight angle and measures $180^{\circ}$.