Triangle interior angles sum to 180°

Dependencies: uniqueness of the parallel through an external point (Through a point off a line, exactly one parallel exists (Playfair)), the converse of "alternate interior angles are equal" (Parallel ⇒ alternate angles equal, denoted B.06), Linear pair sums to 180° (Linear pair sums to 180°).

Statement

Let $\triangle ABC$ be any triangle. Then the sum of its three interior angles is always a straight angle:

This conclusion is independent of the triangle's shape, size, or position — it is the most direct consequence of the axiom "through a point not on a line, there is exactly one parallel" in the Euclidean plane.