What is the reason behind non-horizontal support and resistance lines (wedge, triangle)?

It's a great question and it is utterly, completely mysterious.

Nobody has a clue.

It would be like asking any sociological question .. why is Starbucks popular, why do Westerners use sitdown toilets .. who knows?

You could write any number of guesses at the "reason" but it's just guessing.

An important observation:

• all the things you're mentioning were worked out, noticed, by "Honma, 'God of the markets'" who invented what we now call charting or so-called technical analysis.

He lived in the 1700s and was the first "chartist" or "chart trader" (focussing on what we call "candlesticks" in particular).

(There has been nothing new in charting since then. He noticed all of the four or five patterns that are charting and "technical analysis" - and that's all there is.)

The interesting thing is that Honma back in the day himself pointed out we have not a clue about the "reasons" for these patterns - just that they do exist and (in his opinion) you can, in a word, make money from them.

Once again,

• You are asking for the "reasons" they exist. It is an utter mystery, completely unknown and likely unknowable. (You can trivially find a million pages of writing by idiots guessing at the "reasons".)

• They are completely experiential: they "happen to exist" and (if you "believe" in technical analysis) you "use them" accepting point 1 just above.

• This was all entirely worked out by Honma. (And, he made the observation that the "reasons are unknowable/meaningless".) Nothing has been added since then.

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If (for some reason) you want my particular sheer vague guess at the reasons for these triangle things that some have made piles of money from, my guess is: they are basically geometrical expresses (graphs if you will) of the two human emotions: fear-desperation-panic and elation-idiocy-confusion. So, fear sort of compresses then explodes .. etc. My "theory" is no more thought out or detailed than that, because, see point 1 above - it's completely unknowable, like any sociological, human-mass-movement question.

Regarding as you say how things keep "hitting a slope" (not just a more understandable level line): to me it's just an expression of those two emotions. First the less gutsy (ie: less stupid, less risky) people get shaken out by relatively bigger moves; more and more gutsy/stupid/risky people hang on with "smaller and smaller tolerances". When it busts one way or the other everyone who was wrong gets shaken out instantly and you get the explosion one way or the other. But - that's just a vague look at things, and may be utterly incorrect.

Note too that feedback loops are a huge thing in human behavior. It could be that "angled lines" exist more and more because . . . people just see them, and if you don't stick to them you will lose - because others see them. But again, that's just a trendy thing you hear, could be hopelessly incorrect.

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If you are interested in Honma (sometimes "Honna" etc) and his badass rice trading, can trivially google books etc., eg: amazon.com/dp/B00J24NXJK

Also, the main modern-times source of pretty much all technical trading is the Nison book, which basically represents Honma for today: amazon.com/dp/B00CS74GBM

What are the chances that you could look up into the sky at night and see a cluster of stars that forms a pattern? Perhaps the Big Dipper? Or a W shaped group such as Cassiopeia?

Similarly, if you look at a set of data points such as the price bars of stocks, they form random patterns. Trust me, the trading desk at Goldman Sachs isn't saying, "Let's form a descending triangle with today's trading in Netflix."

And if you stare at a Jackson Pollock painting long enough, you might see the face of your mother-in-law in it.