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      "original": "Which two displayed triangles must be congruent, and why?",
      "stem": "Audit the printed triangle measurements. First explain the supplied intended congruence DEF≅GHI by SAS. Then use the Sine Rule to explain why the displayed whole-degree angles and side lengths must be treated as rounded, rather than a jointly exact metric specification. All printed labels remain in the diagram.",
      "evidence": "For DEF the exact labels would require6sin48°=5sin63° by the Sine Rule. Rigorous interval evaluation gives6sin48°−5sin63°=[0.00383633192252609828963443677741029928176312001, 0.00383633192252609828963443677741029928231413298], strictly positive. The supplied intended69° SAS comparison is preserved, but the rounded diagram is not certified as an exact metric specification."
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      "original": "PC is tangent at C, PD bisects ∠CPE, arc CD=70°, arc DE=50°, and ∠DQE=40°, where Q is the intersection of BD and CE. Find(a)arc BC,(b)∠CPB,(c)arc AB,(d)the arc AE not containing C.",
      "stem": "Audit the printed configuration. Reproduce the supplied formal angle-chasing answers30°,20°,10°,200°, then explain why those formal equations do not make all the source givens jointly realizable. Use the independent coordinate check in the reviewed solution to identify the failed angle-bisector condition. This is a written source-error diagnostic; the source’s numerical answers are retained for comparison.",
      "evidence": "NormalizeC90°,D20°,E330°. The40°intersecting-chord angle forcesB120°in the shown order. The tangent/BD intersection has x=(cos50°−sin70°)/cos70°. A ray fromE making the required−40°angle would meet the tangent at x=√3/2−3/(2tan40°). Rigorous interval evaluation of their difference gives[0.053512807224070780640793465189483044044764266, 0.0535128072240707806407934651894830440453841556], strictly positive. Therefore they are different points: the printed bisector condition cannot hold with the other source givens."
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