# Question 30373: Two Angle-Sum Threshold Defects [Return to catalogue page 152](https://ocme.evemisslab.com/en/questions/page/152/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r01-angle-foundations/#q-20261010-advance-20261014-r01-angle-0073) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0059-30373 · q-20261010-advance-20261014-r01-angle-0073 ## Original question and givens Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=22, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?). - G001 — Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=22, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-angle](https://ocme.evemisslab.com/en/mko/mko-angle/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density14-angle-pair-defects, density14_angle_pair_defects, finite_nondirected_degree_protocol] After derived = [[sum_minus90, sum_minus180, sum_equals90, sum_equals180]] Conditions: none Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Angle ORIGINAL a plus b := sum(22, 0) After Angle ORIGINAL a plus b = 22 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Angle ORIGINAL sum minus90 := sub(22, 90) After Angle ORIGINAL sum minus90 = -68 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Angle ORIGINAL sum minus180 := sub(22, 180) After Angle ORIGINAL sum minus180 = -158 Conditions: none Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [22, 0, 22] After derived = [-68, -158, No, No] Conditions: none Subtract the TWO distinct thresholds90/180 from the SAME sum,and test exact equalities separately.The derived sum may reach360;do not cap it to raw180 or wrap a modulus. Step identifiers and dependenciesRule: Angle_TWO_distinct_threshold_defects_and_exact_equalities · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. - S004 · premise_and_result_audit — Recompute Angle ORIGINAL sum minus180 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Angle ORIGINAL sum minus90 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Angle ORIGINAL a plus b with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate. ## Result (-68, -158, No, No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 9f138ae431f70e2fe33d313c6ea30b7c5b5beb9c41fdf7db90c750ac31fa620f - candidate record SHA-256: 390aedd424ed16f446d77def0381ad0af8e631280a0612a6f9feb70d68e97896 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r01-angle-0073.json)