Sum of Angles of Triangle equals Two Right Angles

Theorem
In a triangle, the sum of the three interior angles equals two right angles.

Proof

 * Triangle With Extension and Parallel.png

Let $\triangle ABC$ be a triangle, and let $BC$ be extended to a point $D$.

Construct $CE$ through the point $C$ parallel to the straight line $AB$.

We have that $AB \parallel CE$ and $AC$ is a transversal that cuts them.

From Parallelism implies Equal Alternate Interior Angles:
 * $\angle BAC = \angle ACE$

Similarly, we have that $AB \parallel CE$ and $BD$ is a transversal that cuts them.

From Parallelism implies Equal Corresponding Angles:
 * $\angle ECD = \angle ABC$

Thus by Euclid's Second Common Notion:
 * $\angle ACD = \angle ABC + \angle BAC$

Again by by Euclid's Second Common Notion:
 * $\angle ACB + \angle ACD = \angle ABC + \angle BAC + \angle ACB$

But from Two Angles on Straight Line make Two Right Angles, $ACB + ACD$ equals two right angles.

So by Euclid's First Common Notion, $\angle ABC + \angle BAC + \angle ACB$ equals two right angles.