Exact Nondirected Angle Budgets and Thresholds
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261010-advance-20261014-r01-angle-foundations · 2026-10-10 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=0, candidate first=0. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 0)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 0).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
2. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=45. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 45)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 45).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
3. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=180, candidate first=30. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 150)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 150).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
4. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=-1. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(No, 91)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (No, 91).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
5. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=100. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(No, -10)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (No, -10).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
6. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=84. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 6)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 6).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
7. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=83. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 7)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 7).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
8. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=82. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 8)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 8).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
9. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=81. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 9)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 9).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
10. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=80. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 10)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 10).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
11. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=79. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 11)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 11).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
12. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=78. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 12)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 12).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
13. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=77. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 13)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 13).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
14. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=76. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 14)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 14).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
15. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=75. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 15)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 15).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
16. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=74. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 16)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 16).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
17. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=73. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 17)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 17).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
18. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=72. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 18)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 18).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
19. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=71. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 19)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 19).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
20. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=70. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 20)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 20).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
21. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=69. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 21)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 21).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
22. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=68. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 22)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 22).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
23. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=67. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 23)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 23).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
24. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=66. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 24)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 24).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
25. Angle-Split Eligibility and Signed Remainder
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. total=90, candidate first=65. Give (is first in [0,total]?, total−first). For an outside candidate still compute the signed arithmetic remainder, without claiming a constructed angle.
View answer and explanation
(Yes, 25)
Check the candidate range before exact subtraction; finite arithmetic is not a ray-measure proof. The exact result, retaining the requested order, is (Yes, 25).
Reviewed step-by-step derivation
Angle · qf-density14-angle-split-candidate · candidate
26. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=0. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 90, 180)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 90, 180).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
27. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=90. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 0, 90)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 0, 90).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
28. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=180. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(No, none, 0)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (No, none, 0).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
29. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=4. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 86, 176)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 86, 176).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
30. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=5. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 85, 175)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 85, 175).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
31. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=6. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 84, 174)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 84, 174).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
32. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=7. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 83, 173)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 83, 173).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
33. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=8. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 82, 172)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 82, 172).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
34. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=9. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 81, 171)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 81, 171).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
35. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=10. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 80, 170)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 80, 170).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
36. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=11. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 79, 169)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 79, 169).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
37. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=12. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 78, 168)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 78, 168).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
38. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=13. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 77, 167)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 77, 167).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
39. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=14. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 76, 166)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 76, 166).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
40. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=15. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 75, 165)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 75, 165).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
41. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=16. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 74, 164)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 74, 164).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
42. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=17. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 73, 163)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 73, 163).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
43. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=18. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 72, 162)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 72, 162).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
44. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=19. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 71, 161)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 71, 161).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
45. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=20. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 70, 160)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 70, 160).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
46. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=21. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 69, 159)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 69, 159).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
47. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=22. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 68, 158)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 68, 158).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
48. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=23. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 67, 157)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 67, 157).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
49. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=24. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 66, 156)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 66, 156).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
50. Complement Eligibility and Supplement
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=25. Give (is a≤90?, eligible complement 90−a or none, supplement 180−a).
View answer and explanation
(Yes, 65, 155)
Guard the complementary angle with the 90° threshold; compute the supplementary angle independently using 180°. The exact result, retaining the requested order, is (Yes, 65, 155).
Reviewed step-by-step derivation
Angle · qf-density14-angle-complement-supplement-pair · candidate
51. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=0, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-90, -180, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-90, -180, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
52. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=45, b=45. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(0, -90, Yes, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (0, -90, Yes, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
53. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=90, b=90. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(90, 0, No, Yes)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (90, 0, No, Yes).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
54. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=180, b=180. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(270, 180, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (270, 180, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
55. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=4, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-86, -176, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-86, -176, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
56. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=5, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-85, -175, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-85, -175, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
57. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=6, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-84, -174, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-84, -174, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
58. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=7, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-83, -173, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-83, -173, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
59. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=8, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-82, -172, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-82, -172, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
60. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=9, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-81, -171, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-81, -171, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
61. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=10, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-80, -170, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-80, -170, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
62. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=11, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-79, -169, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-79, -169, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
63. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=12, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-78, -168, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-78, -168, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
64. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=13, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-77, -167, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-77, -167, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
65. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=14, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-76, -166, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-76, -166, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
66. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=15, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-75, -165, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-75, -165, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
67. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=16, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-74, -164, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-74, -164, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
68. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=17, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-73, -163, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-73, -163, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
69. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=18, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-72, -162, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-72, -162, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
70. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=19, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-71, -161, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-71, -161, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
71. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=20, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-70, -160, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-70, -160, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
72. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=21, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-69, -159, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-69, -159, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
73. Two Angle-Sum Threshold Defects
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?).
View answer and explanation
(-68, -158, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-68, -158, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
74. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=23, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-67, -157, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-67, -157, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
75. Two Angle-Sum Threshold Defects
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. a=24, b=0. Give (a+b−90, a+b−180, is the sum 90?, is the sum 180?).
View answer and explanation
(-66, -156, No, No)
Add the angle values, then compare the two different thresholds separately; do not infer values from a diagram. The exact result, retaining the requested order, is (-66, -156, No, No).
Reviewed step-by-step derivation
Angle · qf-density14-angle-pair-defects · candidate
76. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[0, 0, 0], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([0, 0, 0], Yes, 180)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([0, 0, 0], Yes, 180).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
77. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[90, 90, 0], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([90, 180, 180], Yes, -90)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([90, 180, 180], Yes, -90).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
78. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[90, 90, 90], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([90, 180, 270], No, -90)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([90, 180, 270], No, -90).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
79. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[180, 1, 0], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([180, 181, 181], No, -91)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([180, 181, 181], No, -91).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
80. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[4, 90, 5], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([4, 94, 99], Yes, 81)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([4, 94, 99], Yes, 81).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
81. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[5, 90, 6], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([5, 95, 101], Yes, -11)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([5, 95, 101], Yes, -11).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
82. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[6, 90, 7], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([6, 96, 103], Yes, 77)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([6, 96, 103], Yes, 77).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
83. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[7, 90, 8], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([7, 97, 105], Yes, -15)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([7, 97, 105], Yes, -15).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
84. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[8, 90, 9], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([8, 98, 107], Yes, 73)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([8, 98, 107], Yes, 73).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
85. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[9, 90, 10], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([9, 99, 109], Yes, -19)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([9, 99, 109], Yes, -19).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
86. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[10, 90, 11], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([10, 100, 111], Yes, 69)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([10, 100, 111], Yes, 69).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
87. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[11, 90, 12], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([11, 101, 113], Yes, -23)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([11, 101, 113], Yes, -23).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
88. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[12, 90, 13], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([12, 102, 115], Yes, 65)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([12, 102, 115], Yes, 65).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
89. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[13, 90, 14], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([13, 103, 117], Yes, -27)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([13, 103, 117], Yes, -27).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
90. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[14, 90, 15], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([14, 104, 119], Yes, 61)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([14, 104, 119], Yes, 61).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
91. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[15, 90, 16], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([15, 105, 121], Yes, -31)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([15, 105, 121], Yes, -31).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
92. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[16, 90, 17], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([16, 106, 123], Yes, 57)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([16, 106, 123], Yes, 57).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
93. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[17, 90, 18], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([17, 107, 125], Yes, -35)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([17, 107, 125], Yes, -35).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
94. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[18, 90, 19], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([18, 108, 127], Yes, 53)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([18, 108, 127], Yes, 53).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
95. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[19, 90, 20], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([19, 109, 129], Yes, -39)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([19, 109, 129], Yes, -39).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
96. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[20, 90, 21], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([20, 110, 131], Yes, 49)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([20, 110, 131], Yes, 49).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
97. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[21, 90, 22], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([21, 111, 133], Yes, -43)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([21, 111, 133], Yes, -43).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
98. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[22, 90, 23], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([22, 112, 135], Yes, 45)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([22, 112, 135], Yes, 45).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
99. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[23, 90, 24], target=90. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([23, 113, 137], Yes, -47)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([23, 113, 137], Yes, -47).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
100. Three-Part Prefix Budget and Eligibility
Use exact degrees for nondirected angles in [0,180], not a directed rotation or a measurement from a sketch. Ordered parts=[24, 90, 25], target=180. Give (three prefix sums, is the total≤180?, target−total).
View answer and explanation
([24, 114, 139], Yes, 41)
Accumulate in the original order without taking a modulus; a total above 180 is not an angle in this entry’s nondirected slice. The exact result, retaining the requested order, is ([24, 114, 139], Yes, 41).
Reviewed step-by-step derivation
Angle · qf-density14-angle-three-part-budget · candidate
Canonical question JSONL · English JSONL · Validation report