# Question 5847: Labeled Right Vertex and Opposite Side

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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.

R0010-05847 · q-20261008-advance-20261012-r05-right-triangle-0072



## Original question and givens

Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 22, 68], 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=[90, 22, 68], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).





## 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-right-triangle](https://ocme.evemisslab.com/en/mko/mko-right-triangle/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: exact_right_angle_protocol · Uses: G001





S002 · calculationBefore

three candidate angle sum := sum(90, 22, 68)

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.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

angle sum minus180 := sub(180, 180)

After

angle sum minus180 = 0

Conditions: none

The signed angle-sum defect is total minus180.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

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.

Step identifiers and dependenciesRule: positive_triangle_angle_qualification · Uses: G001, S003





S005 · impliesBefore

original labeled angles=[90, 22, 68];right vertex=1

After

(1, [2, 3], [22, 68])

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.

Step identifiers and dependenciesRule: marked_right_vertex_opposite_endpoints · Uses: G001, S004









## Reverse checks


- 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.





## Result

(1, [2, 3], [22, 68])

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: 8c34a02f8a47658de373f5871eee168d9d6eefc9257aa7db4097adb9a794cfb5
- candidate record SHA-256: 0987cec16697b24c0f75046993d2b44fd285351753bf0b021fd544c69b994c4f
- semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
- question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r05-right-triangle-0072.json)