What’s a proof that the angles of a triangle add up to 180°?

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:

triangle proof

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?

Answer

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).

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Attribution
Source : Link , Question Author : Joe Z. , Answer Author : B H

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