Sum of Angles of Triangle equals Two Right Angles/Proof 2

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Theorem

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


In the words of Euclid:

In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles.

(The Elements: Book $\text{I}$: Proposition $32$)


Proof

TriangleWithLine.png

Let $\Delta ABC$ be a triangle.

Let $DAE$ be a line such that $DE \parallel BC$.

By Parallelism implies Equal Corresponding Angles, $\angle DAB = \angle ABC$ and $\angle EAC = \angle ACB$.

Therefore, the sum of the three angles is $\angle ABC + \angle BCA + \angle CAB = \angle DAB + \angle BAC + \angle CAE = 180^\circ$.

$\blacksquare$