Nebenwinkel ergänzen sich zu ${180}^{\circ }180^\backslash circ$.

Daher ist ${\alpha }^{\ast }={180}^{\circ }-\alpha \backslash alpha^\backslash ast=180^\backslash circ-\backslash alpha$ und genauso ${\beta }^{\ast }={180}^{\circ }-\beta \backslash beta^\backslash ast=180^\backslash circ-\backslash beta$ und ${\gamma }^{\ast }={180}^{\circ }-\gamma \backslash gamma^\backslash ast=180^\backslash circ-\backslash gamma$.

Der Term $({180}^{\circ }-\alpha )+({180}^{\circ }-\beta )+({180}^{\circ }-\gamma )\backslash left(180^\backslash circ-\backslash alpha\backslash right)+\backslash left(180^\backslash circ-\backslash beta\backslash right)+\backslash left(180^\backslash circ-\backslash gamma\backslash right)$ ist also die Summe der Außenwinkel.

Wenn du dies als ${540}^{\circ }-(\alpha +\beta +\gamma )540^\backslash circ-\backslash left(\backslash alpha+\backslash beta+\backslash gamma\backslash right)$ schreibst, und beachtest, dass die Summe der Winkel im Dreieck $\alpha +\beta +\gamma ={180}^{\circ }\backslash alpha+\backslash beta+\backslash gamma=180^\backslash circ$ ist, erhältst du, dass die Summe der Außenwinkel gleich ${540}^{\circ }-{180}^{\circ }={360}^{\circ }540^\backslash circ-180^\backslash circ=360^\backslash circ$ ist.