Exact Angle Qualification and Labeled Right Vertices
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261008-advance-20261012-r05-right-triangle-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=1°. Give (β in degrees, α+β in degrees).
View answer and explanation
(89, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (89, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
2. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=89°. Give (β in degrees, α+β in degrees).
View answer and explanation
(1, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (1, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
3. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=45°. Give (β in degrees, α+β in degrees).
View answer and explanation
(45, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (45, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
4. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=5°. Give (β in degrees, α+β in degrees).
View answer and explanation
(85, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (85, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
5. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=6°. Give (β in degrees, α+β in degrees).
View answer and explanation
(84, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (84, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
6. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=7°. Give (β in degrees, α+β in degrees).
View answer and explanation
(83, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (83, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
7. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=8°. Give (β in degrees, α+β in degrees).
View answer and explanation
(82, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (82, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
8. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=9°. Give (β in degrees, α+β in degrees).
View answer and explanation
(81, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (81, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
9. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=10°. Give (β in degrees, α+β in degrees).
View answer and explanation
(80, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (80, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
10. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=11°. Give (β in degrees, α+β in degrees).
View answer and explanation
(79, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (79, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
11. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=12°. Give (β in degrees, α+β in degrees).
View answer and explanation
(78, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (78, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
12. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=13°. Give (β in degrees, α+β in degrees).
View answer and explanation
(77, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (77, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
13. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=14°. Give (β in degrees, α+β in degrees).
View answer and explanation
(76, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (76, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
14. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=15°. Give (β in degrees, α+β in degrees).
View answer and explanation
(75, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (75, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
15. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=16°. Give (β in degrees, α+β in degrees).
View answer and explanation
(74, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (74, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
16. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=17°. Give (β in degrees, α+β in degrees).
View answer and explanation
(73, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (73, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
17. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=18°. Give (β in degrees, α+β in degrees).
View answer and explanation
(72, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (72, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
18. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=19°. Give (β in degrees, α+β in degrees).
View answer and explanation
(71, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (71, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
19. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=20°. Give (β in degrees, α+β in degrees).
View answer and explanation
(70, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (70, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
20. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=21°. Give (β in degrees, α+β in degrees).
View answer and explanation
(69, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (69, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
21. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=22°. Give (β in degrees, α+β in degrees).
View answer and explanation
(68, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (68, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
22. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=23°. Give (β in degrees, α+β in degrees).
View answer and explanation
(67, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (67, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
23. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=24°. Give (β in degrees, α+β in degrees).
View answer and explanation
(66, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (66, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
24. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=25°. Give (β in degrees, α+β in degrees).
View answer and explanation
(65, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (65, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
25. Acute-Angle Remainder in a Given Right Triangle
Exact degrees: a nondegenerate plane triangle has γ=90° and α=26°. Give (β in degrees, α+β in degrees).
View answer and explanation
(64, 90)
Under the explicit right-angle and triangle-angle-sum premises, subtract α from 90. No tolerance or unstated geometric construction is used. The exact result, retaining the requested order, is (64, 90).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-angle-remainder · candidate
26. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 90, 0]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(No, No, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (No, No, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
27. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[60, 60, 60]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, No, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, No, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
28. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 30, 60]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
29. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[-1, 90, 91]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(No, No, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (No, No, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
30. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 45, 46]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(No, No, 1)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (No, No, 1).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
31. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[0, 0, 180]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(No, No, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (No, No, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
32. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 7, 83]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
33. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 8, 82]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
34. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 9, 81]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
35. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 10, 80]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
36. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 11, 79]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
37. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 12, 78]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
38. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 13, 77]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
39. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 14, 76]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
40. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 15, 75]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
41. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 16, 74]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
42. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 17, 73]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
43. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 18, 72]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
44. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 19, 71]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
45. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 20, 70]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
46. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 21, 69]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
47. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 22, 68]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
48. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 23, 67]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
49. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 24, 66]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
50. Candidate Triangle-Angle Qualification
Exact-degree candidate angles=[90, 25, 65]. Give (all positive with sum 180?, qualified and containing a 90° angle?, signed angle sum−180).
View answer and explanation
(Yes, Yes, 0)
Check positivity and the exact sum before the right angle; zero or negative candidate angles may be rejected even if the sum defect is zero. No constructed triangle is claimed. The exact result, retaining the requested order, is (Yes, Yes, 0).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-validity-pair · candidate
51. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 45, 45], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [45, 45])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [45, 45]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
52. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[1, 90, 89], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [1, 89])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [1, 89]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
53. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[3, 87, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [3, 87])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [3, 87]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
54. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 4, 86], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [4, 86])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [4, 86]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
55. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[5, 90, 85], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [5, 85])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [5, 85]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
56. Labeled Right Vertex and Opposite Side
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).
View answer and explanation
(3, [1, 2], [6, 84])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [6, 84]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
57. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 7, 83], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [7, 83])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [7, 83]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
58. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[8, 90, 82], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [8, 82])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [8, 82]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
59. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[9, 81, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [9, 81])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [9, 81]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
60. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 10, 80], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [10, 80])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [10, 80]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
61. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[11, 90, 79], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [11, 79])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [11, 79]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
62. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[12, 78, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [12, 78])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [12, 78]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
63. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 13, 77], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [13, 77])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [13, 77]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
64. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[14, 90, 76], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [14, 76])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [14, 76]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
65. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[15, 75, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [15, 75])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [15, 75]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
66. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 16, 74], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [16, 74])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [16, 74]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
67. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[17, 90, 73], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [17, 73])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [17, 73]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
68. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[18, 72, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [18, 72])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [18, 72]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
69. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 19, 71], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [19, 71])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [19, 71]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
70. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[20, 90, 70], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [20, 70])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [20, 70]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
71. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[21, 69, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [21, 69])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [21, 69]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
72. Labeled Right Vertex and Opposite Side
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).
View answer and explanation
(1, [2, 3], [22, 68])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [22, 68]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
73. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[23, 90, 67], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(2, [1, 3], [23, 67])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (2, [1, 3], [23, 67]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
74. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[24, 66, 90], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(3, [1, 2], [24, 66])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (3, [1, 2], [24, 66]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
75. Labeled Right Vertex and Opposite Side
Exact degrees: vertex indices 1,2,3 have ordered angles=[90, 25, 65], already required positive, summing to 180, with a unique 90°. Give (right-vertex index, other two endpoint indices, their ordered acute angles).
View answer and explanation
(1, [2, 3], [25, 65])
Find the right angle in label order; the opposite side uses the other two vertex labels. Do not reorder angles or infer a construction proof. The exact result, retaining the requested order, is (1, [2, 3], [25, 65]).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-marked-vertices · candidate
76. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(1/2)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(90/1, Yes)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (90/1, Yes).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
77. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(5/2)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(450/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (450/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
78. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(0/1)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(0/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (0/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
79. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(-1/2)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(-90/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (-90/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
80. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(1/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(180/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (180/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
81. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(2/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(360/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (360/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
82. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(3/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(540/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (540/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
83. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(4/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(720/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (720/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
84. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(5/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(900/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (900/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
85. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(6/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(1080/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (1080/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
86. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(7/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(180/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (180/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
87. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(8/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(1440/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (1440/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
88. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(9/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(1620/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (1620/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
89. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(10/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(1800/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (1800/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
90. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(11/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(1980/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (1980/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
91. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(12/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(2160/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (2160/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
92. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(13/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(2340/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (2340/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
93. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(14/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(360/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (360/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
94. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(15/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(2700/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (2700/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
95. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(16/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(2880/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (2880/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
96. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(17/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(3060/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (3060/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
97. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(18/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(3240/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (3240/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
98. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(19/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(3420/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (3420/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
99. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(20/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(3600/7, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (3600/7, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
100. Exact π-Multiple Candidate without Periodic Wrapping
Candidate θ=(21/7)π radians. Give (exact degree value, does it equal exactly 90°?), without taking a modulus or wrapping a period.
View answer and explanation
(540/1, No)
Multiply the rational π coefficient by 180 and reduce exactly; 450° is not 90°, with no floating-point π or tolerance. The exact result, retaining the requested order, is (540/1, No).
Reviewed step-by-step derivation
Right triangle · qf-density12-right-triangle-pi-multiple · candidate
Canonical question JSONL · English JSONL · Validation report