Triangle Angle Sum Theorem

A proof that the interior angles of any triangle sum to 180°, using a line through one vertex parallel to the opposite side.

Keywords: angle sum, 180 degrees, parallel line, alternate interior angles

Prerequisites: parallel-lines-intro · Difficulty: beginner

The three interior angles of any triangle add up to exactly $180°$:

Strategy: draw a line through vertex $A$ parallel to side $BC$. The alternate interior angles equal $\angle B$ and $\angle C$; together with $\angle A$ they form a straight line ($180°$) at vertex $A$.

Step 1: The Triangle and Its Angles

We start with triangle $ABC$ and mark its three interior angles.

Step 2: Identify Equal Angles

The transversal $AB$ crossing the parallel lines creates alternate interior angles: $\angle B^{\prime }=\angle B$.

Similarly, transversal $AC$ gives $\angle C^{\prime }=\angle C$.

These are the alternate interior angle theorem in action!

Step 3: Three Angles Form a Straight Line

At vertex $A$, the three angles $\angle B^{\prime }$, $\angle A$, and $\angle C^{\prime }$ together form a straight line (the parallel line). A straight angle measures $180°$.

Since $\angle B^{\prime }=\angle B$ and $\angle C^{\prime }=\angle C$:

The three interior angles of any triangle sum to $180°$:

This follows from the fact that alternate interior angles formed by a transversal crossing parallel lines are equal. ∎