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Does anyone have some simple proofs here?

I was looking for some proofs for corresponding angles are equal, but in the one i found they use this theorem that states that the interior angles of two parallel lines (made by the transversal) add up to 180 degrees. And im unfamiliar with this theorem.

excellence
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  • In the proof i found for corresponding angles are equal, they said that the interior angles equal 180 degrees, and to back this up they used something like the euclidian postulate, which gets too far and complicated for me. See it yourself if you want to: – excellence Dec 07 '20 at 11:32
  • https://www.youtube.com/watch?v=-qudm0q5ezQ – excellence Dec 07 '20 at 11:32
  • For the really classical approach, it is Proposition 29, Book 1 of Euclid's elements (see page 32 of this pdf translation). – user854214 Dec 07 '20 at 11:34
  • So there is no easy explanation for this? – excellence Dec 07 '20 at 12:28
  • Does anyone have a video, just something on yt, i cant find any – excellence Dec 07 '20 at 12:30
  • Proving basic plane geometry theorems is a surprisingly thorny subject. There isn't really an accepted axiomatic structure (Elements makes an attempt that was amazing for its time, but doesn't quite cut it these days) for plane geometry, and every mathematical theory needs some kind of axiomatic structure at its base. It's still possible to geometric proofs without such an axiomatic structure, but it requires some results to be held to be true without proof. It's difficult to answer this question without knowing what results you're happy to accept without proof. – user854214 Dec 07 '20 at 13:28
  • One construction could be to drop a perpendicular line to both lines, away from the traversal. This forms a quadrilateral, with two right angles. The angle sum of a quadrilateral is $360^\circ$. Take away the two right angles, we see the cointerior angles sum to $180^\circ$. But this requires assuming angle sums of quadrilaterals (or triangles), that if lines $L_1$ and $L_2$ are parallel and $L_3$ is perpendicular to $L_1$, then it is perpendicular to $L_2$, that there even is a perpendicular line, etc, etc. It's hard for us to know what out of these you'll accept, and what you'll want proven. – user854214 Dec 07 '20 at 13:32

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