The Number of Degrees in a Triangle

It is a given that the sum of the angles in a triangle is \(180^{\circ}\). In passing, it is interesting to note that the proof of this is actually quite easy and within the grasp of sixth graders. Consider the triangle \(ABC\) below. Extend sides \(\overline{AB}\) and \(\overline{CB}\) as shown, and construct a line \(\overline{DBE}\) parallel to the base line \(\overline{AC}\), thereby creating angle \(F\), angle \(G\) and angle \(H\).

By the laws of intersecting lines,
\(\angle F=\angle BAC\text{ (transversal intersecting parallel lines)}\)
\(\angle H=\angle ACB\text{ (transversal intersecting parallel lines)}\)
\(\angle G=\angle ABC\text{ (vertical angles)}\)

Therefore,
\((\angle BAC)+(\angle ACB)+(\angle ABC)=(\angle F)+(\angle H)+(\angle G)\)

Since
\(\angle F)+(\angle H)+(\angle G)=180^{\circ}\)

It follows that
\((\angle BAC)+(\angle ACB)+(\angle ABC)=180^{\circ}\)

John Schwarz