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  "body": "\n## Definition\n\nLet $A$, $B$ and $C$ be points of a [[Euclidean space|0XPR1E85W0]] $\\mathbb{R}^n$ with $A \\neq B$ and $C \\neq B$.\nThe angle $\\angle ABC$ is the [[angle between the vectors|0ZEHZY85E6]] $A - B$ and $C - B$. Thus it is the unique real number $\\theta$ with $0 \\leq \\theta \\leq \\pi$ such that\n$$\\cos \\theta = \\frac{(A - B) \\cdot (C - B)}{\\|A - B\\| \\, \\|C - B\\|}.$$\nThe point $B$ is the vertex of the angle.\n\n## Remarks\n\nThe angle is symmetric: $\\angle ABC = \\angle CBA$.\nThe angle is $\\pi$ if and only if $B$ lies strictly between $A$ and $C$ on a line. It is $0$ if and only if $A$ and $C$ are on the same ray from $B$.\nMathlib's `EuclideanGeometry.angle` uses the same formula for all points. It gives $\\pi / 2$ when $A = B$ or $C = B$.\nThe angle is a right angle when $\\theta = \\pi / 2$, acute when $\\theta \u003c \\pi / 2$ and obtuse when $\\theta \u003e \\pi / 2$.\n\n## Explanations\n\n### L1 human\n\nTo measure the angle at $B$, stand at $B$ and look at $A$, then turn to look at $C$. The angle is how far you turn, using the short way round. The order of $A$ and $C$ does not matter.\n",
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