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 $AB$.

Since $AB \parallel CE$ and $AC$ is a transversal that cuts them, it follows that $\angle BAC = \angle ACE$.

Similarly, since $AB \parallel CE$ and $BD$ is a transversal that cuts them, it follows that $\angle ECD = \angle ABC$.

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