# The Triangle Angle Sum Theorem

Introduction

One of the most elementary concepts we have learned about triangles in Geometry is the angle sum theorem. The theorem states that the sum the three interior angles of a triangle is $180^{o}$. We can easily see this by duplicating or cutting the corners of a triangle and meeting the angles at a particular point (see first figure). The adjacent angles will form a straight angle which is equal to $180^{o}$.

The proof of the angle sum theorem is quite easy. We just need to draw an extra line.

Given

Triangle $ABC$.

Theorem

The interior angle sum of triangle $ABC$ is equal to 180˚.

Proof

Draw a line segment passing through $A$ and parallel to $BC$. This is possible because given a line and a point, you can draw exactly one line through that point parallel to the first line.

Construct points $D$ and $E$ to the left and right of $A$ respectively as shown in the figure below. Now $BC$ and $DE$ are parallel, so $AB$ and $AC$ can be considered as transversals.

Now,

$\angle BAD \cong \angle ABC$ since they are alternate interior angles.

Also, $\angle EAC \cong \angle BCA$ since they are alternate interior angles.

$\angle BAD$, $\angle BAC$, and $\angle EAC$ form a straight angle therefore their sum is $180^{o}$ .

But these three angles are also the interior angles of triangle $ABC$ (Can you see why?)

Therefore, the interior angle sum of triangle $ABC$ is $180^{o}$.