Back in grade school, I had a solution involving “folding the triangle” into a rectangle half the area, and seeing that all the angles met at a point:
However, now that I’m in university, I’m not convinced that this proof is the best one (although it’s still my favourite non-rigorous demonstration). Is there a proof in, say, linear algebra, that the sum of the angles of a triangle is 180 degrees? Or any other Euclidean proofs that I’m not aware of?
Here’s a decent Euclidean proof:
Let x be the line parallel to side AB of △ABC that goes through point C (the line is unique because of the fifth postulate). AC cuts x and AB at the same angle, ∠BAC (corollary of the fifth postulate). BC cuts x and AB at the same angle, ∠ABC. These two angles and the final angle ∠ACB form a straight angle on x, which is always 180∘ (corollary of the third postulate).