{
  "title": "Axioms (Euclid's Geometry) - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "How many axioms did Euclid propose?", "options": {"A": "5", "B": "10", "C": "15", "D": "20"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Euclid's first axiom states:", "options": {"A": "Things equal to same thing are equal", "B": "If equals added to equals, wholes equal", "C": "If equals subtracted from equals, remainders equal", "D": "Things coinciding are equal"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Axioms are:", "options": {"A": "Proved statements", "B": "Self-evident truths", "C": "Theorems", "D": "Postulates"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "Which is Euclid's axiom? 'If equals are added to equals...'", "options": {"A": "The wholes are equal", "B": "The sums are equal", "C": "The results are equal", "D": "The totals are equal"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Euclid's third axiom states:", "options": {"A": "If equals subtracted from equals, remainders equal", "B": "Things equal to same thing are equal", "C": "Whole is greater than part", "D": "Things coinciding are equal"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "'The whole is greater than the part' is which axiom?", "options": {"A": "Fourth", "B": "Fifth", "C": "Third", "D": "Second"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Which is NOT a Euclidean axiom?", "options": {"A": "Things equal to same thing are equal", "B": "Through two points infinite lines can pass", "C": "If equals added to equals, wholes equal", "D": "Whole is greater than part"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "Euclid's fourth axiom states:", "options": {"A": "Things coinciding are equal", "B": "Things equal to same thing are equal", "C": "If equals subtracted from equals, remainders equal", "D": "Whole is greater than part"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If A=B and B=C, then A=C by which axiom?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If 5=5 and 3=3, then 5+3=5+3 by which axiom?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "If line segment AB = CD, and we remove equal parts, remainders equal by axiom:", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "Two things that coincide are equal by axiom:", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "D"},
    {"id": 13, "difficulty": "easy", "question": "A part is always less than whole by axiom:", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fifth"}, "correct_answer": "D"},
    {"id": 14, "difficulty": "easy", "question": "Which axiom justifies: If x=y and y=z, then x=z?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "If AB = PQ and BC = QR, then AB+BC = PQ+QR by axiom:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "Removing equal amounts from equal amounts gives equal remainders by axiom:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 17, "difficulty": "easy", "question": "Superposition principle is related to axiom:", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "D"},
    {"id": 18, "difficulty": "easy", "question": "The statement 'All right angles are equal' is:", "options": {"A": "Axiom", "B": "Postulate", "C": "Theorem", "D": "Definition"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "How many postulates did Euclid give?", "options": {"A": "3", "B": "5", "C": "7", "D": "9"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "Difference between axiom and postulate:", "options": {"A": "Axioms for arithmetic, postulates for geometry", "B": "No difference", "C": "Axioms more general", "D": "Postulates more general"}, "correct_answer": "C"},

    {"id": 21, "difficulty": "medium", "question": "Using Euclid's axioms, prove that if 2x=2y, then x=y.", "options": {"A": "Use axiom 3 (subtraction)", "B": "Use axiom 1 (transitivity)", "C": "Use axiom 2 (addition)", "D": "Use axiom 4 (coincidence)"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "Which axiom justifies: If AB=CD and EF=GH, then AB+EF=CD+GH?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Prove: If a=b and c=d, then a-c=b-d.", "options": {"A": "Axiom 1 and 3", "B": "Axiom 2 and 4", "C": "Axiom 3 only", "D": "Axiom 5 only"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "'Things which are double of the same thing are equal to each other.' This follows from:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "If ∠A=∠B and ∠B=∠C, then ∠A=∠C by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Which Euclid's axiom is used in solving equation x+3=7 by subtracting 3?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "Prove that if a=b, then -a=-b.", "options": {"A": "Use axiom 3 with c=-a", "B": "Use axiom 2 with c=-a", "C": "Use axiom 1", "D": "Use axiom 5"}, "correct_answer": "A"},
    {"id": 28, "difficulty": "medium", "question": "If two line segments coincide exactly, they are equal by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "'The whole is greater than the part' implies:", "options": {"A": "Part can equal whole", "B": "Part always less than whole", "C": "Sometimes part equals whole", "D": "Part can be greater than whole"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "In triangle, sum of two sides > third side. This is:", "options": {"A": "Axiom", "B": "Postulate", "C": "Theorem", "D": "Definition"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "Which Euclid's axiom justifies: If a=b and b=c, then a=c?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If AB=CD, and we add EF to both, AB+EF=CD+EF by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "To prove that if x+a=y+a then x=y, we use:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "The method of superposition in geometry relies on:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "D"},
    {"id": 35, "difficulty": "medium", "question": "Which axiom prevents a part from being equal to whole?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fifth"}, "correct_answer": "D"},
    {"id": 36, "difficulty": "medium", "question": "If A=B, B=C, C=D, then A=D by repeated use of:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Prove: If 2x=2y, then x=y. Which axiom?", "options": {"A": "Use axiom 3 with subtraction", "B": "Use axiom 2 with addition", "C": "Use axiom 1", "D": "Use axiom 5"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "'Things which are halves of the same thing are equal.' Proved using:", "options": {"A": "Axiom 1", "B": "Axiom 3", "C": "Axiom 4", "D": "Axiom 5"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "In algebraic identity (a+b)²=a²+2ab+b², which axiom is fundamental?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "If two triangles are congruent, their areas are equal. This uses:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "D"},
    {"id": 41, "difficulty": "medium", "question": "Prove that if a=b and c=d, then ac=bd.", "options": {"A": "Uses axioms 1,2,3", "B": "Uses axioms 2 repeatedly", "C": "Uses axioms 3 repeatedly", "D": "Uses axiom 4"}, "correct_answer": "B"},
    {"id": 42, "difficulty": "medium", "question": "Which axiom is used in solving 2x+3=11?", "options": {"A": "Axiom 3 twice", "B": "Axiom 2 twice", "C": "Axiom 1 and 3", "D": "Axiom 4 and 5"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "If ∠A+∠B=∠C+∠D and ∠A=∠C, then ∠B=∠D by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "medium", "question": "'All right angles are equal' is Euclid's:", "options": {"A": "Axiom 4", "B": "Postulate 4", "C": "Common notion", "D": "Postulate 3"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "If a line segment is divided into two equal parts, each part is less than whole by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 5"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "medium", "question": "Prove that if a=b, then a²=b².", "options": {"A": "Use axiom 2 with a and b", "B": "Use axiom 2 repeatedly", "C": "Use axiom 3", "D": "Use axiom 4"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "medium", "question": "In proving SAS congruence, which axiom is used?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "If two circles have equal radii, they are congruent by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "D"},
    {"id": 49, "difficulty": "medium", "question": "Which Euclid's axiom is the basis for transitivity of equality?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "medium", "question": "If a=b, then 3a=3b by:", "options": {"A": "Axiom 1", "B": "Axiom 2 applied twice", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "medium", "question": "Prove that if a+c=b+c, then a=b.", "options": {"A": "Use axiom 3", "B": "Use axiom 2", "C": "Use axiom 1", "D": "Use axiom 4"}, "correct_answer": "A"},
    {"id": 52, "difficulty": "medium", "question": "In geometry, 'things which coincide with one another are equal' allows:", "options": {"A": "Superposition method", "B": "Transitive property", "C": "Addition property", "D": "Subtraction property"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "If AB=2 cm and CD=2 cm, then AB=CD by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "medium", "question": "To prove midpoint divides segment into two equal parts, we use:", "options": {"A": "Axiom 1 and 4", "B": "Axiom 2 and 3", "C": "Axiom 3 and 5", "D": "Axiom 4 and 5"}, "correct_answer": "D"},
    {"id": 55, "difficulty": "medium", "question": "Which axiom justifies: If a=b and c is any number, then a+c=b+c?", "options": {"A": "First", "B": "Second", "C": "Third", "D": "Fourth"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "medium", "question": "If two angles are supplements of equal angles, they are equal. Proof uses:", "options": {"A": "Axiom 1 and 3", "B": "Axiom 2 and 3", "C": "Axiom 3 only", "D": "Axiom 4 only"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "In proving triangle inequality theorem, axiom 5 is used to show:", "options": {"A": "Sum > part", "B": "Transitivity", "C": "Addition property", "D": "Coincidence"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "medium", "question": "If a=b, then a/2=b/2 (for nonzero 2) by:", "options": {"A": "Axiom 3", "B": "Axiom 2", "C": "Axiom 1", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "medium", "question": "Which axiom prevents Russell's paradox in set theory?", "options": {"A": "Axiom 5", "B": "Axiom 1", "C": "Axiom 3", "D": "Not covered by Euclid"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "medium", "question": "Prove that if a=b and c=d, then a/c=b/d (c,d≠0).", "options": {"A": "Uses axiom 3", "B": "Uses axiom 2", "C": "Uses axioms 1,2,3", "D": "Cannot be proved from Euclid's axioms alone"}, "correct_answer": "D"},
    {"id": 61, "difficulty": "medium", "question": "In coordinate geometry, distance formula is derived using:", "options": {"A": "Euclid's axioms", "B": "Pythagoras theorem", "C": "Both", "D": "Neither"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "If two rectangles have equal length and width, they are congruent by:", "options": {"A": "Axiom 4", "B": "Axiom 2", "C": "Axiom 1", "D": "Axiom 3"}, "correct_answer": "A"},
    {"id": 63, "difficulty": "medium", "question": "Which axiom is used to prove that vertical angles are equal?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "C"},
    {"id": 64, "difficulty": "medium", "question": "Prove that if a=b, then -a=-b.", "options": {"A": "Use axiom 3 with 0", "B": "Use axiom 2 with -a", "C": "Use axiom 1", "D": "Use axiom 4"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "'The whole is greater than the part' implies set A is proper subset of B then:", "options": {"A": "n(A)<n(B)", "B": "A⊂B", "C": "B has more elements", "D": "All of these"}, "correct_answer": "D"},
    {"id": 66, "difficulty": "medium", "question": "Which Euclid's axiom corresponds to additive identity?", "options": {"A": "None directly", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "If a=b, then a²+2ab+b²=b²+2ab+a² by:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 68, "difficulty": "medium", "question": "Prove that if (a-b)=(c-d) and b=d, then a=c.", "options": {"A": "Use axiom 3", "B": "Use axiom 2", "C": "Use axiom 1", "D": "Use axiom 4"}, "correct_answer": "B"},
   {"id": 69, "difficulty": "medium", "question": "In proving angles in linear pair are supplementary, we use:", "options": {"A": "Axiom 2", "B": "Axiom 3", "C": "Axiom 4", "D": "Postulate 5"}, "correct_answer": "D"},
    {"id": 70, "difficulty": "medium", "question": "Which axiom justifies method of elimination in algebra?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "C"},
    {"id": 71, "difficulty": "medium", "question": "If two squares have equal sides, they are congruent by:", "options": {"A": "Axiom 4", "B": "Axiom 1", "C": "Axiom 2", "D": "Axiom 3"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Prove that if a/b=c/d and b=d≠0, then a=c.", "options": {"A": "Use axiom 3", "B": "Use axiom 2", "C": "Use axiom 1", "D": "Cannot be proved from Euclid's axioms alone"}, "correct_answer": "D"},
    {"id": 73, "difficulty": "medium", "question": "In Euclidean geometry, parallel postulate is:", "options": {"A": "Axiom", "B": "Postulate", "C": "Theorem", "D": "Definition"}, "correct_answer": "B"},
    {"id": 74, "difficulty": "medium", "question": "Which axiom is used in proving isosceles triangle theorem?", "options": {"A": "Axiom 4", "B": "Axiom 1", "C": "Axiom 3", "D": "Axiom 5"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "If a=b, then a³=b³ by:", "options": {"A": "Repeated use of axiom 2", "B": "Axiom 1", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "Prove that if a+x=b+x, then a=b.", "options": {"A": "Use axiom 3", "B": "Use axiom 2", "C": "Use axiom 1", "D": "Use axiom 4"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "In proving SSS congruence, we use:", "options": {"A": "Axiom 4", "B": "Axiom 1", "C": "Axiom 2", "D": "Axiom 3"}, "correct_answer": "A"},
    {"id": 78, "difficulty": "medium", "question": "'Equals added to equals give equals' is:", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 4"}, "correct_answer": "B"},
    {"id": 79, "difficulty": "medium", "question": "Which axiom prevents a line segment from being equal to its proper part?", "options": {"A": "Axiom 1", "B": "Axiom 2", "C": "Axiom 3", "D": "Axiom 5"}, "correct_answer": "D"},
    {"id": 80, "difficulty": "medium", "question": "Prove that if 2a=2b, then a=b.", "options": {"A": "Use axiom 3 with a", "B": "Use axiom 3 twice", "C": "Use axiom 2", "D": "Use axiom 1"}, "correct_answer": "A"},

    {"id": 81, "difficulty": "hard", "question": "Using Euclid's axioms, prove that if a=b and c=d, then a-c=b-d.", "options": {"A": "Use axiom 3 directly", "B": "Use axiom 2 and 3", "C": "Use axiom 1 and 3", "D": "Use axiom 4 and 5"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "Prove that sum of angles of triangle is 180°. Which Euclid elements are used?", "options": {"A": "Axioms and postulates", "B": "Axioms only", "C": "Postulates only", "D": "Definitions only"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Show that Euclid's axioms are insufficient for proving a(b+c)=ab+ac.", "options": {"A": "True, need distributive property", "B": "False, can be proved", "C": "Sometimes true", "D": "Depends on context"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "Prove uniqueness of parallel line through point not on given line. Uses:", "options": {"A": "Parallel postulate", "B": "Axiom 1", "C": "Axiom 5", "D": "All axioms"}, "correct_answer": "A"},
    {"id": 85, "difficulty": "hard", "question": "Using Euclid's axioms, prove that if a>b and b>c, then a>c.", "options": {"A": "Use axiom 1 and 5", "B": "Use axiom 2 and 5", "C": "Use axiom 3 and 5", "D": "Cannot be proved from axioms alone"}, "correct_answer": "D"},
    {"id": 86, "difficulty": "hard", "question": "Prove that vertical angles are equal using Euclid's elements.", "options": {"A": "Use axiom 1,3 and postulate 4", "B": "Use axiom 2 only", "C": "Use axiom 5 only", "D": "Use postulate 5 only"}, "correct_answer": "A"},
    {"id": 87, "difficulty": "hard", "question": "Show that Euclid's fifth axiom is independent of first four.", "options": {"A": "By non-Euclidean geometries", "B": "Cannot be shown", "C": "By counterexample", "D": "By contradiction"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "Prove that base angles of isosceles triangle are equal.", "options": {"A": "Use SAS congruence and axiom 4", "B": "Use axiom 1 only", "C": "Use axiom 2 only", "D": "Use axiom 5 only"}, "correct_answer": "A"},
    {"id": 89, "difficulty": "hard", "question": "Using Euclid's axioms, prove √2 is irrational.", "options": {"A": "Cannot be proved with axioms alone", "B": "Use axiom 3", "C": "Use axiom 5", "D": "Use all axioms"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "Prove that diagonals of parallelogram bisect each other.", "options": {"A": "Use ASA congruence and axioms", "B": "Use axiom 2 only", "C": "Use axiom 3 only", "D": "Use axiom 4 only"}, "correct_answer": "A"},
    {"id": 91, "difficulty": "hard", "question": "Show that Euclid's axioms are consistent.", "options": {"A": "By constructing Cartesian model", "B": "Cannot be shown", "C": "By contradiction", "D": "By induction"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Prove Pythagorean theorem using Euclid's elements.", "options": {"A": "Uses similar triangles and axioms", "B": "Uses axiom 1 only", "C": "Uses axiom 5 only", "D": "Cannot be proved"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Using axioms, prove that if a/b=c/d and b=d≠0, then a=c.", "options": {"A": "Requires multiplicative property not in axioms", "B": "Use axiom 3", "C": "Use axiom 2", "D": "Use axiom 1"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Prove that angle in semicircle is right angle.", "options": {"A": "Uses triangle angle sum and axioms", "B": "Uses axiom 2 only", "C": "Uses axiom 3 only", "D": "Uses axiom 5 only"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "Show that Euclid's system is incomplete without betweenness axioms.", "options": {"A": "True, missing continuity", "B": "False, complete", "C": "Sometimes complete", "D": "Never complete"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Prove that opposite sides of parallelogram are equal.", "options": {"A": "Use ASA congruence and axiom 4", "B": "Use axiom 1 only", "C": "Use axiom 2 only", "D": "Use axiom 3 only"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Using axioms, prove that if a<b and c<d, then a+c<b+d.", "options": {"A": "Cannot be proved from Euclid's axioms", "B": "Use axiom 2", "C": "Use axiom 3", "D": "Use axiom 5"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "Prove that three angle bisectors of triangle are concurrent.", "options": {"A": "Uses angle bisector theorem and axioms", "B": "Uses axiom 1 only", "C": "Uses axiom 2 only", "D": "Uses axiom 5 only"}, "correct_answer": "A"},
    {"id": 99, "difficulty": "hard", "question": "Show that Euclid's first four axioms hold in spherical geometry but fifth doesn't.", "options": {"A": "True", "B": "False", "C": "First four also fail", "D": "All hold"}, "correct_answer": "C"},
    {"id": 100, "difficulty": "hard", "question": "Prove that there are infinitely many prime numbers using Euclid's method.", "options": {"A": "Uses contradiction and divisibility", "B": "Uses axiom 1 only", "C": "Uses axiom 3 only", "D": "Cannot be proved from geometric axioms"}, "correct_answer": "D"}
  ]
}