{
  "aliases": [
    "triangle-angle-sum-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0NTSE7EREA"
    },
    {
      "field": "uses",
      "target": "0NZVJYZVTZ"
    }
  ],
  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in a Euclidean space $\\mathbb{R}^n$, and let $\\angle$ denote the [[angle at a point|0GJAJC6TNZ]]. Then\n$$\\angle BAC + \\angle CBA + \\angle ACB = \\pi.$$\n\n## Remarks\n\nMathlib proves the formula under the weaker hypothesis that $B \\neq A$ and $C \\neq A$, with its convention for the angle at a repeated point.\nAn equilateral triangle has three equal angles, so each one is $\\pi / 3$.\nIn the axioms of plane geometry, the statement is equivalent to the parallel postulate. It is false in spherical geometry, where the sum is more than $\\pi$, and in hyperbolic geometry, where it is less.\n\n## History\n\nEuclid's Elements proves the statement as part of Proposition 32 of Book I.\n\n## Explanations\n\n### L1 human\n\nCut a paper triangle and tear off its three corners. Put the corners side by side with their tips at one point. They always fill a straight line, which is half a turn, or $\\pi$ radians.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "aliases": [
      "triangle-angle-sum-theorem"
    ],
    "kind": "statement",
    "msc": [
      "51M04"
    ],
    "name": "Angle sum of a triangle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7840113",
      "wikipedia": "Sum of angles of a triangle"
    },
    "sources": [
      {
        "locator": "Euclidean geometry",
        "url": "https://en.wikipedia.org/wiki/Sum_of_angles_of_a_triangle"
      }
    ],
    "tag": "05MDKQBEAF",
    "uses": [
      "0CTNFGYMM2",
      "0GJAJC6TNZ"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/05MDKQBEAF/",
  "links": [
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    },
    {
      "field": "uses",
      "target": "0GJAJC6TNZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/05MDKQBEAF.md",
  "msc": [
    "51M04"
  ],
  "name": "Angle sum of a triangle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/05M/05MDKQBEAF.md",
  "status": null,
  "tag": "05MDKQBEAF",
  "uri": "m2://05MDKQBEAF",
  "url": "https://math2.tamnd.com/wiki/angle-sum-of-a-triangle/",
  "writing": {
    "tag": "05MDKQBEAF",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}