{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-angle",
  "semantic_family_id": "sf-angle",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "角",
    "en": "Angle"
  },
  "summary": {
    "zh-Hant": "在初等歐幾里得幾何中，角由具有共同端點的兩條射線形成，並可用度數描述其張開量。"
  },
  "statement": {
    "zh-Hant": "兩條具有共同端點的射線形成一個角；共同端點稱為頂點，兩條射線稱為角的邊，本物件以度數表示 0° 到 180° 的非定向角。"
  },
  "formula": {
    "tex": "\\theta",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>θ</mi></math>",
    "semantic_ast": {
      "type": "symbol",
      "name": "theta",
      "glyph": "θ"
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2b4af44cb17349bb9452292a447fe70d48c9ce08e68dd2f74e9628cf8ac625b8"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "採初等歐幾里得幾何與度數量值，且只處理 0° 到 180° 的非定向角。"
    }
  ],
  "symbols": [
    {
      "token": "θ",
      "role": "angle_or_measure",
      "role_zh": "角或其明示量值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-point-line-plane",
      "title_zh": "點、線、面",
      "reason_zh": "角的頂點與兩條射線使用基礎幾何對象語言。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "角的大小由兩條邊之間的張開量決定，不由射線畫得多長或多粗決定。0°<θ<90°、θ=90°、90°<θ<180°、θ=180° 的角，分別稱為銳角、直角、鈍角與平角。",
      "兩角量值和為 90° 時互餘，和為 180° 時互補；這些關係依賴量值相加，不等於兩角必須在圖上相鄰。",
      "本物件先固定非定向初等角。定向旋轉、同終邊角、弧度與內積空間角需要額外的方向、週期或代數結構。"
    ],
    "common_misconception_zh": "常見誤解是由示意圖外觀猜角度、把角的邊長當成角的大小，或認為互餘與互補角必須相鄰。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "明示度數可以做有限算術重算，但不構成角、射線與度量性質的形式證明。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "選定 EuclideanGeometry 或內積空間模型，精確綁定射線、頂點、角量值與度數轉換。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-angle.md 整理；待人工審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-09-22"
  }
}