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:::I'm not sure what your point is with spherical trigonometry, the angles are still rotations and the sets of rotations around a point forms a circle group. The fact that the rotations form SO(3) and can be labelled with dihedral angles or 3D bivectors is interesting but not really relevant to defining angles.
:::As far as writing for a non-technical audience, [[WP:TECHNICAL]] is of course the guide. But notably there is [[WP:OVERSIMPLIFY]]: "Encyclopedia articles should not 'tell lies to children'". So dumbing down the definition of angle is not the right approach. I would say the lead paragraph should give a good definition in terms of the circle group and define general concepts, and then a second paragraph can then go on to describe common examples. Lie groups are relevant to defining angles in higher dimensions, but the basic notion of angle is defined in a plane, so the Lie groups can go in a section. [[User:Mathnerd314159|Mathnerd314159]] ([[User talk:Mathnerd314159|talk]]) 18:33, 5 May 2023 (UTC)
::::Element of the circle group is too reductive (as you say, we should not [[WP:OVERSIMPLIFY]]). First, [[circle group]] explicitly describes such elements as complex numbers (not just as elements of an abstract group defined structurally with multiple possible representations), when the most popular representation of angles is as angle measures in a periodic interval. Second, this only covers angles in a particular plane, without allowing the possibility for angles to include orientation. Third, this does not allow for contexts where angle measure per se does not make sense, for example in referring to an angle in an affine space, or an angle in a space where the coordinates are elements of a finite field. Fourth, this does not allow for angles that can spin around many times, as seen in e.g. [[angular velocity]]. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:43, 5 May 2023 (UTC)
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