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


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.