What are the angles used in angle connectors and liftarms? LEGO makes Technic Angle Connectors, like these:








(Here are names and part numbers).

They also make angular liftarms, like these:

 (part 32348).

That last part has an angle of 53.5 degrees.  What are the angles of the angle connectors, and why were these angles chosen?  (I'd understand angles like 45, 60, and 72 degrees, but 53.5?!)
The regular connector have simple angles which are multiples of 22.5°. #4 is 45°, #3 is 22.5° and #5 is 67.5°. That's pretty boring stuff, but there you go.

The 53.5 angle isn't as bizarre as you would think and there's a very good reason for it (and less boring to me, but YMMV).

That reason is that it forms a Pythagorean triangle, and specifically the best known 3:4:5 one, meaning these liftarms can be used together with regular right-angle frames to make sturdy constructions.

The one you chose isn't the best example as the beam doesn't extend to the full  hypotenuse (the connection points need to be 5 studs apart), but the longer one is perfect for that. If the small side is attached to a beam, the cross at the end of the long side will be perfectly aligned on a beam that would be 4 studs higher.

LEGO uses that to connect beams 4 studs apart in a very solid way, for example in 8435 - 4WD (the blue beam behind the front wheel) or in the more recent 8053 - Mobile Crane (left of the linear actuator)

And for the record, the angle in degrees is weirder that just 53.5°; it's actually 53.130102...°, or the invert cosine of 3/5. So if you thought Pythagoras and trigonometry wouldn't come and annoy you in your play, you're oh so wrong.