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R0010-05831 · q-20261008-advance-20261012-r05-right-triangle-0056
Exact degrees: vertex indices 1,2,3 have ordered angles=[6, 84, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
G001 — Exact degrees: vertex indices 1,2,3 have ordered angles=[6, 84, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).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.
Before
angle unit=degrees;exact declared protocol
After
check qualifications before geometric conclusions;no tolerance
Conditions: none
Use the declared exact angle protocol,not a sketch or floating tolerance.Candidate angle data do not by themselves construct a triangle.
Rule: exact_right_angle_protocol · Uses: G001
Before
three candidate angle sum := sum(6, 84, 90)
After
three candidate angle sum = 180
Conditions: none
Add the three ordered candidate angles completely;negative or zero values remain data to audit,not assumed valid triangle angles.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
angle sum minus180 := sub(180, 180)
After
angle sum minus180 = 0
Conditions: none
The signed angle-sum defect is total minus180.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
positive flags=[Yes, Yes, Yes];sum=180;defect=0
After
positive180-degree qualification=Yes
Conditions: none
The triangle-angle protocol requires all angles positive and sum exactly180;zero sum defect does not excuse a zero or negative angle.
Rule: positive_triangle_angle_qualification · Uses: G001, S003
Before
original labeled angles=[6, 84, 90];right vertex=3
After
(3, [1, 2], [6, 84])
Conditions: none
Use the original1-based right-vertex index;the opposite-side endpoints are the other two vertices.Do not sort angles and thereby relabel vertices.
Rule: marked_right_vertex_opposite_endpoints · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute angle sum minus180 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute three candidate angle sum with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(3, [1, 2], [6, 84])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 03989612213351ddfa9788e7a945766a6873722ae224cce6cd90b7c2e106780dcandidate record SHA-256: d0fa1cdf7985fb857c2dba53de70998e033a0659529b54854ae3f60e3161a387semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED