# Question 5808: Acute-Angle Remainder in a Given Right Triangle

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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.

R0010-05808 · q-20261008-advance-20261012-r05-right-triangle-0008



## Original question and givens

Exact degrees: a nondegenerate plane triangle has γ=90° and α=9°. Give (β in degrees, α+β in degrees).


- G001 — Exact degrees: a nondegenerate plane triangle has γ=90° and α=9°. Give (β in degrees, α+β in degrees).





## 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

acute-angle budget := sub(180, 90)

After

acute-angle budget = 90

Conditions: none

Under the supplied nondegenerate Euclidean plane-triangle premises,the180-degree sum minus the90-degree right angle leaves90 for the two acute angles.

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





S003 · calculationBefore

other acute angle beta := sub(90, 9)

After

other acute angle beta = 81

Conditions: none

Subtract the supplied α from the two-acute-angle budget.

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





S004 · calculationBefore

alpha plus beta := sum(9, 81)

After

alpha plus beta = 90

Conditions: none

Add back to audit the acute-angle sum,retaining the requested field order.

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





S005 · impliesBefore

1≤α=9≤89;β=81

After

α+β=90;both acute

Conditions: none

The strict acute range for α also places β between1 and89 degrees.This derives values under the given right-triangle premises,not a construction proof.

Step identifiers and dependenciesRule: right_triangle_remainder_premises · 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 — Recompute alpha plus beta with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute other acute angle beta with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute acute-angle budget 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

(81, 90)

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: 6a878b2b328c8ef1b78e95006f02a4e504872041279ae136c6c0c18e18de8b88
- candidate record SHA-256: a13e50a069c8bee9148431b9d861566c2563f2c24a1c0c8479e8bcaec5e62acd
- 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-0008.json)