{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-angle-01",
  "mko_id": "mko-angle",
  "title": {
    "zh-Hant": "角：對頂角相等的逐步證明",
    "en": "Angle: a step-by-step proof that vertical angles are equal"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-angle-01",
    "source": {
      "mko_id": "mko-angle",
      "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
      "sha256": "e77eca47694501f576639c69b49c366d6053b453055167a92e61e9a9b88626da",
      "english_sha256": "c5cbf3ced06829082efdaaf6f2d97bf231e72f882e785bc6b4ecdf390768cc8b"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "eq_refl_symm_trans",
      "eq_congruence",
      "add_inverse",
      "numeric_evaluation"
    ],
    "goal": {
      "zh-Hant": "直線 AB 與直線 CD 是兩條相異直線，相交於 O，且 O 在 A、B 之間，也在 C、D 之間。證明對頂角相等：m∠AOC=m∠BOD 且 m∠COB=m∠DOA；再以 m∠AOC=35° 為例算出四個角。",
      "en": "Distinct lines AB and CD meet at O, with O between A and B and between C and D. Prove that vertical angles are equal: m∠AOC=m∠BOD and m∠COB=m∠DOA; then, for m∠AOC=35°, compute all four angles."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "初等平面歐幾里得幾何；角以度數量測，非定向，取值在 0° 到 180°（MKO 假設 a1）。",
          "en": "Elementary plane Euclidean geometry; angles are measured in degrees, unoriented, with values from 0° to 180° (MKO assumption a1)."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "直線 AB 與直線 CD 相異，相交於 O；O 在 A、B 之間，也在 C、D 之間。因兩直線只交於 O，C、D 不在直線 AB 上，A、B 不在直線 CD 上。",
          "en": "Lines AB and CD are distinct and meet at O; O lies between A and B and between C and D. Since the lines meet only at O, C and D are not on line AB, and A and B are not on line CD."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "例子：m∠AOC=35°。",
          "en": "Example: m∠AOC=35°."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2"
        ],
        "before_tex": "\\angle AOC,\\ \\angle COB,\\ \\angle BOD,\\ \\angle DOA",
        "after_tex": "\\alpha=m\\angle AOC,\\ \\beta=m\\angle COB,\\ \\gamma=m\\angle BOD,\\ \\delta=m\\angle DOA",
        "relation": "definition",
        "rule": "def_intro",
        "conditions": [],
        "explanation": {
          "zh-Hant": "為四個以 O 為頂點的角的度數取名。依 MKO，角的大小是兩邊之間的張開量，與射線畫多長無關。∠AOC 與 ∠BOD 是一組對頂角（OA 與 OB 方向相反，OC 與 OD 方向相反），∠COB 與 ∠DOA 是另一組。",
          "en": "Name the measures of the four angles with vertex O. By the MKO, the size of an angle is the opening between its sides, independent of how long the rays are drawn. ∠AOC and ∠BOD are one pair of vertical angles (OA is opposite to OB and OC is opposite to OD); ∠COB and ∠DOA are the other pair."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "def_intro",
            "mko_angle_def"
          ],
          "explanation": {
            "zh-Hant": "此步只取名，不推出新事實。四個角都有定義：例如 C 不在直線 AB 上，所以 OA 與 OC 不共線，∠AOC 是真正的角。",
            "en": "This step only introduces names and derives no new fact. All four angles are defined: for example C is not on line AB, so OA and OC are not collinear and ∠AOC is a genuine angle."
          }
        },
        "leaf_basis": "def_intro + mko_angle_def"
      },
      {
        "id": "S02",
        "uses": [
          "S01",
          "G2"
        ],
        "before_tex": "A\\text{-}O\\text{-}B,\\ C\\notin\\overleftrightarrow{AB}",
        "after_tex": "\\alpha+\\beta=180",
        "relation": "theorem_application",
        "rule": "linear_pair",
        "conditions": [
          "O\\ \\text{between}\\ A,B",
          "C\\notin\\overleftrightarrow{AB}"
        ],
        "explanation": {
          "zh-Hant": "對直線 AB 與點 C 使用線性對公設（X=C）：m∠AOC+m∠COB=180°，即 α+β=180。公設的兩個前提「O 在 A、B 之間」與「C 不在直線 AB 上」都由 G2 提供。",
          "en": "Apply the linear pair postulate to line AB and the point C (X=C): m∠AOC+m∠COB=180°, that is α+β=180. Both hypotheses of the postulate, O between A and B and C not on line AB, come from G2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2",
            "linear_pair"
          ],
          "explanation": {
            "zh-Hant": "公設要求 A、O、B 共線且 O 在中間，以及 C 不在該直線上；兩者都在 G2 中明示。只憑兩角「相鄰」不足以推出和為 180°。",
            "en": "The postulate needs A, O, B collinear with O in the middle, and C off that line; both are stated in G2. Being adjacent alone would not give a sum of 180°."
          }
        },
        "leaf_basis": "linear_pair"
      },
      {
        "id": "S03",
        "uses": [
          "S01",
          "G2"
        ],
        "before_tex": "C\\text{-}O\\text{-}D,\\ B\\notin\\overleftrightarrow{CD}",
        "after_tex": "\\beta+\\gamma=180",
        "relation": "theorem_application",
        "rule": "linear_pair",
        "conditions": [
          "O\\ \\text{between}\\ C,D",
          "B\\notin\\overleftrightarrow{CD}"
        ],
        "explanation": {
          "zh-Hant": "對直線 CD 與點 B 使用線性對公設（公設中的 A、B 換成 C、D，X=B）：m∠COB+m∠BOD=180°，即 β+γ=180。",
          "en": "Apply the linear pair postulate to line CD and the point B (the postulate's A, B become C, D, and X=B): m∠COB+m∠BOD=180°, that is β+γ=180."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2",
            "linear_pair"
          ],
          "explanation": {
            "zh-Hant": "O 在 C、D 之間，且 B 不在直線 CD 上（兩直線相異、只交於 O），都由 G2 給出。",
            "en": "O between C and D, and B not on line CD (the lines are distinct and meet only at O), are both given by G2."
          }
        },
        "leaf_basis": "linear_pair + angle_symmetric"
      },
      {
        "id": "S04",
        "uses": [
          "S02",
          "S03"
        ],
        "before_tex": "\\alpha+\\beta=180,\\ \\beta+\\gamma=180",
        "after_tex": "\\alpha+\\beta=\\gamma+\\beta",
        "relation": "implies",
        "rule": "eq_refl_symm_trans",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩式右邊都是 180，依等號的對稱與傳遞得 α+β=β+γ；再依加法交換律寫成 α+β=γ+β，讓相同的一項 β 位於兩邊同一位置。",
          "en": "Both right sides are 180, so symmetry and transitivity of equality give α+β=β+γ; commutativity of addition rewrites this as α+β=γ+β, with the common term β in the same place on both sides."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S02",
            "S03",
            "eq_refl_symm_trans"
          ],
          "explanation": {
            "zh-Hant": "只需兩個等式；不宣稱反向，即不由 α+β=γ+β 推回兩者各自等於 180。",
            "en": "Only the two equations are needed; the converse, recovering that each sum is 180 from α+β=γ+β, is not claimed."
          }
        },
        "leaf_basis": "eq_refl_symm_trans + add_comm"
      },
      {
        "id": "S05",
        "uses": [
          "S04"
        ],
        "before_tex": "\\alpha+\\beta=\\gamma+\\beta",
        "after_tex": "\\alpha=\\gamma",
        "relation": "equivalent",
        "rule": "L_cancel",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依引理 L_cancel（u=α、v=γ、w=β）消去兩邊的 β，得 α=γ，即 m∠AOC=m∠BOD。",
          "en": "By Lemma L_cancel (u=α, v=γ, w=β) cancel β from both sides to get α=γ, that is m∠AOC=m∠BOD."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "L_cancel"
          ],
          "explanation": {
            "zh-Hant": "反向：兩邊同加 β 回到 α+β=γ+β；引理的兩個方向都已在子步證明。",
            "en": "Backward: add β to both sides to return to α+β=γ+β; both directions of the lemma are proved in the sub-steps."
          }
        },
        "leaf_basis": "eq_congruence + add_assoc + add_inverse + add_zero"
      },
      {
        "id": "S05.1",
        "uses": [
          "eq_congruence"
        ],
        "before_tex": "u+w=v+w",
        "after_tex": "(u+w)+(-w)=(v+w)+(-w)",
        "relation": "equivalent",
        "rule": "eq_congruence",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩邊同加 −w。反向是兩邊同加 w，再用 S05.2 的化簡。",
          "en": "Add −w to both sides. The backward move is to add w to both sides and simplify as in S05.2."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "eq_congruence",
            "add_inverse"
          ],
          "explanation": {
            "zh-Hant": "加 −w 與加 w 互為反運算，對任何實數 w 都可做。",
            "en": "Adding −w and adding w undo each other and are possible for every real w."
          }
        },
        "leaf_basis": "eq_congruence + add_inverse"
      },
      {
        "id": "S05.2",
        "uses": [
          "S05.1"
        ],
        "before_tex": "(u+w)+(-w)=(v+w)+(-w)",
        "after_tex": "u=v",
        "relation": "equivalent",
        "rule": "eq_replace",
        "conditions": [],
        "explanation": {
          "zh-Hant": "化簡左邊：(u+w)+(−w)=u+(w+(−w))=u+0=u；右邊同理得 v。S05.1 與 S05.2 合起來證得引理 L_cancel。",
          "en": "Simplify the left side: (u+w)+(−w)=u+(w+(−w))=u+0=u; the right side gives v in the same way. S05.1 and S05.2 together prove Lemma L_cancel."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "add_assoc",
            "add_inverse",
            "add_zero"
          ],
          "explanation": {
            "zh-Hant": "所用的是對所有實數成立的恆等式，換掉相等的式子不改變等式的真假，雙向成立。",
            "en": "These are identities for all reals; replacing equal expressions does not change whether the equation holds, so the step is two-way."
          }
        },
        "leaf_basis": "add_assoc + add_inverse + add_zero"
      },
      {
        "id": "S06",
        "uses": [
          "S01",
          "G2"
        ],
        "before_tex": "A\\text{-}O\\text{-}B,\\ D\\notin\\overleftrightarrow{AB}",
        "after_tex": "\\delta+\\gamma=180",
        "relation": "theorem_application",
        "rule": "linear_pair",
        "conditions": [
          "O\\ \\text{between}\\ A,B",
          "D\\notin\\overleftrightarrow{AB}"
        ],
        "explanation": {
          "zh-Hant": "對直線 AB 與點 D 使用線性對公設（X=D）：m∠AOD+m∠DOB=180°。依 angle_symmetric，m∠AOD=m∠DOA=δ，m∠DOB=m∠BOD=γ，所以 δ+γ=180。",
          "en": "Apply the linear pair postulate to line AB and the point D (X=D): m∠AOD+m∠DOB=180°. By angle_symmetric, m∠AOD=m∠DOA=δ and m∠DOB=m∠BOD=γ, so δ+γ=180."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G2",
            "linear_pair",
            "angle_symmetric"
          ],
          "explanation": {
            "zh-Hant": "需要 O 在 A、B 之間，且 D 不在直線 AB 上，兩者都由 G2 給出；改寫角名稱只用到非定向角與書寫順序無關。",
            "en": "It needs O between A and B and D off line AB, both from G2; renaming the angles only uses that an unoriented angle does not depend on the order of its sides."
          }
        },
        "leaf_basis": "linear_pair + angle_symmetric"
      },
      {
        "id": "S07",
        "uses": [
          "S03",
          "S06"
        ],
        "before_tex": "\\beta+\\gamma=180,\\ \\delta+\\gamma=180",
        "after_tex": "\\beta+\\gamma=\\delta+\\gamma",
        "relation": "implies",
        "rule": "eq_refl_symm_trans",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩式右邊都是 180，依等號的對稱與傳遞得 β+γ=δ+γ。",
          "en": "Both right sides are 180, so symmetry and transitivity of equality give β+γ=δ+γ."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S03",
            "S06",
            "eq_refl_symm_trans"
          ],
          "explanation": {
            "zh-Hant": "與 S04 相同，只用兩個等式，不宣稱反向。",
            "en": "As in S04, only the two equations are used and no converse is claimed."
          }
        },
        "leaf_basis": "eq_refl_symm_trans"
      },
      {
        "id": "S08",
        "uses": [
          "S07"
        ],
        "before_tex": "\\beta+\\gamma=\\delta+\\gamma",
        "after_tex": "\\beta=\\delta",
        "relation": "equivalent",
        "rule": "L_cancel",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依引理 L_cancel（w=γ）消去 γ，得 β=δ，即 m∠COB=m∠DOA。",
          "en": "By Lemma L_cancel (w=γ) cancel γ to get β=δ, that is m∠COB=m∠DOA."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "L_cancel"
          ],
          "explanation": {
            "zh-Hant": "反向：兩邊同加 γ；引理兩向都已證。",
            "en": "Backward: add γ to both sides; both directions of the lemma are proved."
          }
        },
        "leaf_basis": "eq_congruence + add_assoc + add_inverse + add_zero"
      },
      {
        "id": "S09",
        "uses": [
          "S05",
          "S08"
        ],
        "before_tex": "\\alpha=\\gamma,\\ \\beta=\\delta",
        "after_tex": "m\\angle AOC=m\\angle BOD,\\ m\\angle COB=m\\angle DOA",
        "relation": "implies",
        "rule": "def_unfold",
        "conditions": [],
        "explanation": {
          "zh-Hant": "把名稱展開回角：兩組對頂角分別相等。證明沒有用到任何具體度數或圖形量測，所以對每個滿足 G2 的位置都成立。",
          "en": "Unfold the names back to the angles: both pairs of vertical angles are equal. The proof used no particular measure and no measurement of a figure, so it holds for every configuration satisfying G2."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S05",
            "S08",
            "def_unfold"
          ],
          "explanation": {
            "zh-Hant": "目標 G：兩組對頂角相等。足夠條件 H：α=γ 且 β=δ；H⇒G 由 S01 的命名。H 由 S05、S08 從 G2 與公設正向證成，沒有把 G 當作前提。",
            "en": "Goal G: both pairs of vertical angles are equal. Sufficient condition H: α=γ and β=δ; H⇒G by the naming in S01. H is proved forwards from G2 and the postulate in S05 and S08; G is never used as a premise."
          }
        },
        "leaf_basis": "def_unfold + and_intro_elim"
      },
      {
        "id": "S10",
        "uses": [
          "G3",
          "S02",
          "S05",
          "S08"
        ],
        "before_tex": "\\alpha=35",
        "after_tex": "\\beta=145,\\ \\gamma=35,\\ \\delta=145",
        "relation": "implies",
        "rule": "eq_congruence",
        "conditions": [],
        "explanation": {
          "zh-Hant": "代入 α=35：由 S02，β=180−35=145；由 S05，γ=α=35；由 S08，δ=β=145。核對：35+145=180。",
          "en": "Substitute α=35: from S02, β=180−35=145; from S05, γ=α=35; from S08, δ=β=145. Check: 35+145=180."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G3",
            "S02",
            "S05",
            "S08"
          ],
          "explanation": {
            "zh-Hant": "這只是一個例子的計算，可以重算（N1、N2），但不能取代 S02 到 S09 的一般證明。",
            "en": "This is the computation for one example and can be recomputed (N1, N2), but it cannot replace the general proof in S02 to S09."
          }
        },
        "leaf_basis": "eq_congruence + eq_replace + numeric_evaluation"
      }
    ],
    "result": {
      "zh-Hant": "在 G2 的設定下，m∠AOC=m∠BOD 且 m∠COB=m∠DOA：對頂角相等。若 m∠AOC=35°，四個角依序為 35°、145°、35°、145°。",
      "en": "Under the setting of G2, m∠AOC=m∠BOD and m∠COB=m∠DOA: vertical angles are equal. If m∠AOC=35°, the four angles are 35°, 145°, 35° and 145° in order."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "∠AOC 與 ∠COB 有共同邊 OC，所以 m∠AOC+m∠COB=180°",
          "en": "∠AOC and ∠COB share the side OC, so m∠AOC+m∠COB=180°"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "還需要 A、O、B 共線且 O 在中間；三射線方向 0°、50°、120° 給出 50°+70°=120°。",
          "en": "A, O, B must also be collinear with O in the middle; rays at 0°, 50° and 120° give 50°+70°=120°."
        }
      },
      {
        "claim": {
          "zh-Hant": "對頂角相等，所以相鄰的兩角也相等",
          "en": "Vertical angles are equal, so adjacent angles are equal too"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "α=35 時 β=145，相鄰兩角不等。",
          "en": "With α=35 we get β=145, so adjacent angles differ."
        }
      },
      {
        "claim": {
          "zh-Hant": "互補的兩角必須在圖上相鄰",
          "en": "Supplementary angles must be adjacent in the figure"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "30° 與 150° 可在不同圖形中，量值和為 180° 即互補（MKO）。",
          "en": "Angles of 30° and 150° may lie in different figures; their measures sum to 180°, so they are supplementary (MKO)."
        }
      },
      {
        "claim": {
          "zh-Hant": "在圖上量得 ∠AOC 約等於 ∠BOD，所以對頂角相等得證",
          "en": "In a drawing ∠AOC measures about the same as ∠BOD, so the vertical-angle theorem is proved"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "一次量測只支持那張圖並帶有誤差，不涵蓋所有位置。",
          "en": "One measurement supports only that drawing, with error, and does not cover all configurations."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "eq_refl_symm_trans",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "等號的性質：u=u；若 u=v 則 v=u；若 u=v 且 v=w 則 u=w。",
          "en": "Properties of equality: u=u; if u=v then v=u; if u=v and v=w then u=w."
        }
      },
      {
        "id": "eq_congruence",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "等式同用：若 u=v，對兩邊做同一個有定義的運算 f，得 f(u)=f(v)。只對一邊做運算不屬於此規則。",
          "en": "Congruence of equality: if u=v, applying the same defined operation f to both sides gives f(u)=f(v). Operating on one side only is not an instance of this rule."
        }
      },
      {
        "id": "add_inverse",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "加法反元素：每個 u 都有 −u，使 u+(−u)=0；u−v 定義為 u+(−v)。",
          "en": "Additive inverse: every u has −u with u+(−u)=0; u−v means u+(−v)."
        }
      },
      {
        "id": "numeric_evaluation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "明確整數或有理數算式的精確求值；可以獨立重算。",
          "en": "Exact evaluation of an explicit integer or rational expression; independently recomputable."
        }
      },
      {
        "id": "def_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "引入定義：可為一個命題、式子或物件取名；之後名稱與被定義者可互相替換。",
          "en": "Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable."
        }
      },
      {
        "id": "add_comm",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "加法交換律：u+v=v+u。",
          "en": "Commutativity of addition: u+v=v+u."
        }
      },
      {
        "id": "add_assoc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "加法結合律：(u+v)+w=u+(v+w)。",
          "en": "Associativity of addition: (u+v)+w=u+(v+w)."
        }
      },
      {
        "id": "add_zero",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "加法單位元：u+0=u。",
          "en": "Additive identity: u+0=u."
        }
      },
      {
        "id": "eq_replace",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "等值代換：命題中的一個式子可換成與它相等的式子，命題的真假不變。",
          "en": "Replacement of equals: an expression inside a statement may be replaced by an equal expression without changing whether the statement is true."
        }
      },
      {
        "id": "def_unfold",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "展開定義：把一個已定義的詞換成它的定義內容（或反向收回），命題的真假不變。",
          "en": "Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true."
        }
      },
      {
        "id": "and_intro_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "合取：由 P 與 Q 得 P∧Q；由 P∧Q 可分別得 P 與 Q。",
          "en": "Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q."
        }
      },
      {
        "id": "mko_angle_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-angle",
          "sha256": "e77eca47694501f576639c69b49c366d6053b453055167a92e61e9a9b88626da",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "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。",
          "en": "Two rays with a common endpoint form an angle. The common endpoint is the vertex and the rays are the sides; this object treats unoriented angles from 0° through 180° measured in degrees."
        }
      },
      {
        "id": "linear_pair",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "線性對公設：若 O 在 A、B 之間，且點 X 不在直線 AB 上，則 m∠AOX+m∠XOB=180°。",
          "en": "Linear pair postulate: if O lies between A and B and a point X is not on line AB, then m∠AOX+m∠XOB=180°."
        }
      },
      {
        "id": "angle_symmetric",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "非定向角的度數與兩邊的書寫順序無關：m∠XOY=m∠YOX。",
          "en": "The measure of an unoriented angle does not depend on the order in which its sides are written: m∠XOY=m∠YOX."
        }
      },
      {
        "id": "L_cancel",
        "kind": "lemma",
        "proved_in": [
          "S05.1",
          "S05.2"
        ],
        "statement": {
          "zh-Hant": "對實數 u、v、w：u+w=v+w ⇔ u=v。",
          "en": "For reals u, v, w: u+w=v+w ⇔ u=v."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N1",
        "step": "S10",
        "lhs": "180-35",
        "rhs": "145"
      },
      {
        "id": "N2",
        "step": "S10",
        "lhs": "35+145",
        "rhs": "180"
      },
      {
        "id": "N3",
        "lhs": "50+70",
        "relation": "!=",
        "rhs": "180"
      },
      {
        "id": "N4",
        "lhs": "30+150",
        "rhs": "180"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "在 MKO 的範圍內工作：初等平面歐幾里得幾何、度數量值、0° 到 180° 的非定向角。以線性對公設為前提證明對頂角定理；不推導角度公設本身，不處理定向角或弧度；不改動 canonical MKO 或英文 edition。",
      "en": "Works inside the MKO's scope: elementary plane Euclidean geometry, degree measure, unoriented angles from 0° to 180°. Proves the vertical-angle theorem from the linear pair postulate; it does not derive the angle postulates themselves, does not treat directed angles or radians, and does not change the canonical MKO or its English edition."
    },
    "md_symbols": [
      [
        "m\\angle AOC",
        "角 AOC 的度數",
        "the degree measure of angle AOC"
      ],
      [
        "\\alpha,\\beta,\\gamma,\\delta",
        "四個角的度數名稱",
        "names for the four measures"
      ],
      [
        "O",
        "交點，四個角的共同頂點",
        "the intersection point, common vertex of the four angles"
      ],
      [
        "A\\text{-}O\\text{-}B",
        "O 在 A 與 B 之間，三點共線",
        "O lies between A and B on one line"
      ],
      [
        "\\overleftrightarrow{AB}",
        "通過 A、B 的直線",
        "the line through A and B"
      ]
    ],
    "md_error_controls": [
      {
        "title": "量圖不是證明 / Measuring a figure is not a proof",
        "zh": "MKO 指出常見誤解是由示意圖外觀猜角度。在一張圖上量得 ∠AOC 與 ∠BOD 看起來相等，只支持那張圖，而且量角器有誤差；本稿的證明只用 G2 的位置關係與線性對公設，適用於所有滿足 G2 的位置。",
        "en": "The MKO notes the common mistake of guessing measures from a sketch. Measuring ∠AOC and ∠BOD as apparently equal in one drawing supports only that drawing, with measurement error; the proof here uses only the incidence data of G2 and the linear pair postulate, so it covers every configuration satisfying G2."
      },
      {
        "title": "相鄰角不一定互補 / Adjacent angles are not always supplementary",
        "zh": "線性對公設需要 A、O、B 共線且 O 在中間。若從 O 出發的三條射線方向是 0°、50°、120°，相鄰的兩個角是 50° 與 70°，和為 120°，不是 180°。所以 S02、S03、S06 都必須引用 G2 的「O 在兩點之間」。",
        "en": "The linear pair postulate needs A, O, B collinear with O in the middle. If three rays from O point in the directions 0°, 50° and 120°, the two adjacent angles are 50° and 70°, summing to 120°, not 180°. That is why S02, S03 and S06 must cite 'O between the two points' from G2."
      },
      {
        "title": "互補不必相鄰 / Supplementary angles need not be adjacent",
        "zh": "MKO 說明互補只依量值和為 180°，不要求兩角在圖上相鄰。本證明用的是線性對這個特殊的相鄰情況來得到互補；反過來，互補的兩角不必構成線性對。",
        "en": "The MKO explains that being supplementary depends only on the measures summing to 180° and does not require the angles to be adjacent. This proof uses the special adjacent case of a linear pair to obtain supplementary angles; conversely, supplementary angles need not form a linear pair."
      }
    ],
    "md_open_items": [
      {
        "zh-Hant": "線性對公設與 angle_symmetric 是本稿的前提，不在本稿中由更基本的公理推出。",
        "en": "The linear pair postulate and angle_symmetric are premises of this guide; they are not derived here from more basic axioms."
      }
    ]
  },
  "revision": {
    "id": "production-20261019-adopted-96",
    "source_commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
    "adopted_source_sha256": "4966ea2d56ae93cd8d358cb3ea94cb1a4d98d2b97d71a4d59cc9fe07c477d0aa",
    "original_source_sha256": "4966ea2d56ae93cd8d358cb3ea94cb1a4d98d2b97d71a4d59cc9fe07c477d0aa",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "independent_AI_concurrence": "NOT_MEASURED",
    "issue_ids": [],
    "scope": "Owner-adopted natural-language teaching guide; not a formal proof, canonical MKO rewrite or evidence-status promotion."
  }
}