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.

Let $BC$ be extended to a point $D$.

From External Angle of Triangle equals Sum of other Internal Angles:
 * $\angle ACD = \angle ABC + \angle BAC$

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.