三角形的内角和

The sum of the interior angles in a triangle is $180^\circ$. You can prove this by setting one of the vertices of the triangle onto a line parallel to the base. Now think about it: you have 2 transversals intersecting the parallel lines. What do you know about the angles that the transversal makes with the parallel lines? Yes, that's correct: alternate interior angles are congruent. So, you now have 2 alternate interior angles, congruent to 2 of the angles of the triangle. What do you do with this info? What's the third angle on the line? Wait a moment, it's the third angle of the triangle! Wow, so it's proven: the sum of the interior angles in a triangle is $180^\circ$.

As we know, the sum of the interior angles $\alpha$, $\beta$, and $\gamma$ is $180^\circ$. Our exterior angles are $180-\alpha$, $180-\beta$, and $180-\gamma$.

Now, there's only one more step: Getting rid of the $\alpha$, $\beta$, and $\gamma$. But I gave you an equation involving their sum in the very first sentence.