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 $latex 180^{o}$. 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 $latex 180^{o}$.

trianlge angle sum

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

Given

Triangle $latex ABC$.

Theorem

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

Proof

Draw a line segment passing through $latex A$ and parallel to $latex BC$. 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 $latex D$ and $latex E$ to the left and right of $latex A$ respectively as shown in the figure below. Now $latex BC$ and $latex DE$ are parallel, so $latex AB$ and $latex AC$ can be considered as transversals.

triangle angle sum proof

Now,

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

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

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

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

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

 

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