{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\Pi$ be an [[absolute plane|VKFPCSRNDH]] that satisfies the axiom of Archimedes.\nThen the sum of the three [[angles|0GJAJC6TNZ]] of every [[triangle|0CTNFGYMM2]] in $\\Pi$ is at most two right angles.\n\n## Remarks\n\nThe proof starts from a triangle $ABC$ and builds a new triangle with the same angle sum, one of whose angles is at most half of the angle at $A$.\nRepeating this $n$ times gives a triangle with the same angle sum and an angle at most $2^{-n}$ times the angle at $A$.\nIf the angle sum exceeded two right angles by some angle $\\varepsilon$, the axiom of Archimedes would give a step where this small angle is less than $\\varepsilon$.\nThen the other two angles would sum to more than two right angles, which contradicts the [[exterior angle theorem|0NTSE7EREA]].\nThe axiom of Archimedes is needed: Max Dehn built non-Archimedean geometries in which the other axioms hold and some triangles have angle sum greater than two right angles.\n\n## Explanations\n\n### L1 human\n\nWithout the parallel postulate, one cannot prove that the angles of a triangle add up to $180^\\circ$.\nBut one can still prove that they never add up to more than $180^\\circ$.\nIn the Euclidean plane the sum is exactly $180^\\circ$, and in the hyperbolic plane it is always less.\n\n### L2 human\n\nThe theorem is one of the main results of absolute geometry.\nWith it, the parallel postulate is equivalent to the existence of a single triangle with angle sum $180^\\circ$.\nThe defect of a triangle is two right angles minus its angle sum.\nIt is additive when a triangle is cut into two, and in the hyperbolic plane it is proportional to the area.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "VKFPCSRNDH"
    ],
    "kind": "statement",
    "msc": [
      "51F05",
      "51M09"
    ],
    "name": "Saccheri–Legendre theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7396621",
      "wikipedia": "Saccheri–Legendre theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Saccheri%E2%80%93Legendre_theorem"
      }
    ],
    "tag": "1Y2XK0QJDX",
    "uses": [
      "VKFPCSRNDH",
      "0CTNFGYMM2",
      "0GJAJC6TNZ",
      "0NTSE7EREA"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/1Y2XK0QJDX/",
  "links": [
    {
      "field": "about",
      "target": "VKFPCSRNDH"
    },
    {
      "field": "uses",
      "target": "VKFPCSRNDH"
    },
    {
      "field": "uses",
      "target": "0CTNFGYMM2"
    },
    {
      "field": "uses",
      "target": "0GJAJC6TNZ"
    },
    {
      "field": "uses",
      "target": "0NTSE7EREA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1Y2XK0QJDX.md",
  "msc": [
    "51F05",
    "51M09"
  ],
  "name": "Saccheri–Legendre theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1Y2/1Y2XK0QJDX.md",
  "status": null,
  "tag": "1Y2XK0QJDX",
  "uri": "m2://1Y2XK0QJDX",
  "url": "https://tamnd.github.io/math2/wiki/saccheri-legendre-theorem/",
  "writing": {
    "tag": "1Y2XK0QJDX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}