{
  "title": "Theorems on Parallelograms (Rectilinear Figures) - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "If both pairs of opposite sides of quadrilateral are equal, it is a: 🔄", "options": {"A": "Trapezium", "B": "Parallelogram", "C": "Kite", "D": "Rhombus"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "In parallelogram, diagonals: ✖️", "options": {"A": "Are equal", "B": "Bisect each other", "C": "Are perpendicular", "D": "Bisect angles"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "If one pair of opposite sides is equal and parallel, quadrilateral is: 🔄", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Parallelogram", "D": "Square"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "A quadrilateral is parallelogram if both pairs of opposite angles are: 📐", "options": {"A": "Acute", "B": "Equal", "C": "Supplementary", "D": "Right"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "Diagonals of rectangle: ⬛", "options": {"A": "Are perpendicular", "B": "Are equal", "C": "Bisect angles", "D": "Are not equal"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If diagonals of quadrilateral bisect each other, it is: 🔄", "options": {"A": "Parallelogram", "B": "Trapezium", "C": "Kite", "D": "Cyclic"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "In rhombus, diagonals bisect each other at: 💎", "options": {"A": "Acute angles", "B": "Right angles", "C": "Obtuse angles", "D": "180°"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "If in parallelogram, one angle is 90°, then it is: 📏", "options": {"A": "Rhombus", "B": "Rectangle", "C": "Square", "D": "Kite"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "A square has properties of both rectangle and: ⬛", "options": {"A": "Trapezium", "B": "Kite", "C": "Rhombus", "D": "None"}, "correct_answer": "C"},
    {"id": 10, "difficulty": "easy", "question": "In parallelogram, consecutive angles are: 🔄", "options": {"A": "Equal", "B": "Complementary", "C": "Supplementary", "D": "Right"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "If midpoints of sides of quadrilateral are joined, shape formed is: 🔶", "options": {"A": "Square", "B": "Rectangle", "C": "Parallelogram", "D": "Rhombus"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "A quadrilateral with diagonals bisecting at 90° and equal is: ✖️", "options": {"A": "Rectangle", "B": "Square", "C": "Rhombus", "D": "Kite"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "In parallelogram ABCD, if AB=8 cm, then CD= ? 📏", "options": {"A": "4 cm", "B": "8 cm", "C": "12 cm", "D": "16 cm"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "If both pairs of opposite angles are equal in quadrilateral, then it is: 🔄", "options": {"A": "Rectangle", "B": "Parallelogram", "C": "Rhombus", "D": "Trapezium"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "A parallelogram with equal diagonals is: ⚖️", "options": {"A": "Rhombus", "B": "Rectangle", "C": "Square", "D": "Both B and C"}, "correct_answer": "D"},
    {"id": 16, "difficulty": "easy", "question": "In a parallelogram, diagonals divide it into how many congruent triangles? 🔺", "options": {"A": "2", "B": "4", "C": "6", "D": "8"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "If one diagonal of parallelogram bisects one angle, then parallelogram is: 💎", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Square", "D": "Kite"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "The line joining midpoints of two sides of triangle is parallel to third side and equal to: 🔺", "options": {"A": "Half of it", "B": "One third", "C": "Equal", "D": "Double"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "A quadrilateral with one pair of opposite sides equal and parallel is: 📐", "options": {"A": "Rectangle", "B": "Parallelogram", "C": "Trapezium", "D": "Kite"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "If diagonals of parallelogram are perpendicular, it is: ⬛", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Square", "D": "Both B and C"}, "correct_answer": "D"},
    {"id": 21, "difficulty": "medium", "question": "In parallelogram ABCD, E and F are midpoints of AB and CD. Prove DE=BF using which theorem? 📐", "options": {"A": "SSS", "B": "SAS", "C": "ASA", "D": "RHS"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Diagonals of parallelogram are 16 and 20 cm. They intersect at 60°. Find sides if ratio 3:4. ⚖️", "options": {"A": "6,8", "B": "9,12", "C": "12,16", "D": "15,20"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "If in parallelogram ABCD, ∠A = (3x+10)° and ∠B = (2x+30)°, find ∠C. 🔢", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "D"},
    {"id": 24, "difficulty": "medium", "question": "Prove quadrilateral is parallelogram if its diagonals bisect each other. Which axiom is used? 🧠", "options": {"A": "SAS congruence", "B": "ASA congruence", "C": "SSS congruence", "D": "AAS congruence"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "In rhombus, diagonals are 24 and 10 cm. Find perimeter. 💎", "options": {"A": "52", "B": "56", "C": "60", "D": "64"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "ABCD is parallelogram. P on AB, Q on CD such that AP=CQ. Prove PB=DQ using which test? 🔄", "options": {"A": "Opposite sides equal", "B": "SAS", "C": "AAS", "D": "SSS"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "In rectangle, diagonals intersect at O. If ∠AOB=100°, find ∠OBC. ⬛", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Parallelogram with vertices (1,2),(4,3),(3,6),(0,5) has area? 📊", "options": {"A": "6", "B": "8", "C": "10", "D": "12"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "If one angle of parallelogram is 30° more than twice adjacent angle, find angles. 🔢", "options": {"A": "50°,130°", "B": "60°,120°", "C": "70°,110°", "D": "80°,100°"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "In parallelogram, altitude corresponding to side 15 is 8, other side 20, find other altitude. 📏", "options": {"A": "6", "B": "7", "C": "8", "D": "9"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "ABCD parallelogram, AE ⟂ DC, AF ⟂ BC. If AB=10, AE=6, AD=8, find AF. 📐", "options": {"A": "4.8", "B": "5", "C": "6", "D": "7.5"}, "correct_answer": "D"},
    {"id": 32, "difficulty": "medium", "question": "Diagonals AC and BD of parallelogram intersect at O. If AO=3x-1, OC=x+5, find AC. ✖️", "options": {"A": "12", "B": "16", "C": "20", "D": "24"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "Midpoints of parallelogram's sides joined in order form which shape? 🔶", "options": {"A": "Parallelogram", "B": "Rectangle", "C": "Rhombus", "D": "Square"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "In parallelogram, sum of squares of sides equals sum of squares of diagonals. This is called: 📐", "options": {"A": "Apollonius theorem", "B": "Pythagoras theorem", "C": "Midpoint theorem", "D": "Euler's theorem"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "If parallelogram and rectangle have same base and between same parallels, ratio of areas is: 📊", "options": {"A": "1:1", "B": "1:2", "C": "2:1", "D": "Depends on height"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "Diagonals of quadrilateral divide it into 4 triangles of equal area. Quadrilateral is: 🔄", "options": {"A": "Rectangle", "B": "Parallelogram", "C": "Rhombus", "D": "Square"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "In rhombus ABCD, diagonal AC=16, BD=30. Find side length. 💎", "options": {"A": "17", "B": "18", "C": "19", "D": "20"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If both diagonals of quadrilateral bisect each other at right angles, quadrilateral is: ✖️", "options": {"A": "Rectangle", "B": "Rhombus", "C": "Square", "D": "Both B and C"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "Parallelogram with sides 13 and 15, one diagonal 14. Find other diagonal. ⚖️", "options": {"A": "16", "B": "18", "C": "20", "D": "22"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "In parallelogram, point P inside such that area(APB)+area(CPD) = area(ABCD)/2. Where is P? 🎯", "options": {"A": "On diagonal AC", "B": "On diagonal BD", "C": "Midpoint", "D": "Anywhere"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "hard", "question": "In parallelogram ABCD, ∠A=60°, AB=10, AD=6. Find diagonal BD. 📐", "options": {"A": "√76", "B": "√84", "C": "√94", "D": "√104"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "hard", "question": "ABCD parallelogram. E on BC, F on AD such that BE=DF. Prove AECF is parallelogram using which theorem? 🧠", "options": {"A": "Opposite sides equal and parallel", "B": "Diagonals bisect", "C": "Both pairs opposite sides equal", "D": "Any of these"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Parallelogram and rectangle have same perimeter 40. Sides of parallelogram 12 and 8, height 5. Ratio of areas? 📊", "options": {"A": "5:6", "B": "4:5", "C": "3:4", "D": "2:3"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "If midpoints of sides of rectangle are joined, shape formed is: 🔲", "options": {"A": "Square", "B": "Rectangle", "C": "Rhombus", "D": "Parallelogram"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "hard", "question": "In parallelogram, distances from point on diagonal to two adjacent sides are 4 and 6. If sides 10 and 15, find area. 📊", "options": {"A": "60", "B": "75", "C": "90", "D": "120"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "hard", "question": "Diagonals of parallelogram meet at O. P is any point. Prove PA²+PB²+PC²+PD² = OA²+OB²+OC²+OD² + 4OP². This is: 📐", "options": {"A": "Apollonius theorem", "B": "Parallelogram law", "C": "Euler's theorem", "D": "Midpoint theorem"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "hard", "question": "Parallelogram ABCD. Lines through A and C parallel to diagonals BD and AC form parallelogram. Its area compared to ABCD is? 📊", "options": {"A": "Same", "B": "Half", "C": "Double", "D": "Four times"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "hard", "question": "In rhombus, side=10, one diagonal=12. Distance between opposite sides? 💎", "options": {"A": "9.6", "B": "8.8", "C": "7.2", "D": "6.4"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "ABCD parallelogram. P on AB, Q on CD, AP:CQ=2:3. Line PQ meets diagonal AC at R. AR:RC=? 🔄", "options": {"A": "2:3", "B": "3:2", "C": "1:1", "D": "Depends on shape"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "Parallelogram with sides a,b and diagonals d₁,d₂ satisfies: ⚖️", "options": {"A": "d₁²+d₂²=2(a²+b²)", "B": "d₁²-d₂²=4ab", "C": "d₁d₂=a²+b²", "D": "d₁/d₂=a/b"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "If E,F are midpoints of sides BC,AD of parallelogram ABCD, prove AECF is parallelogram using which test? 📐", "options": {"A": "Opposite sides equal", "B": "One pair equal and parallel", "C": "Diagonals bisect", "D": "Any of these"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "hard", "question": "In parallelogram, sum of squares of all sides=100. Sum of squares of diagonals=? 📐", "options": {"A": "50", "B": "100", "C": "150", "D": "200"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "hard", "question": "From parallelogram ABCD, line through D meets BC produced at E. AB=8, AD=5, EC=3. Find BE. 🔄", "options": {"A": "10", "B": "11", "C": "12", "D": "13"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "Point inside parallelogram divides it into 4 triangles of areas w,x,y,z in order. Then: 📊", "options": {"A": "w+x=y+z", "B": "w+y=x+z", "C": "w+z=x+y", "D": "w=x, y=z"}, "correct_answer": "B"},
    {"id": 55, "difficulty": "hard", "question": "If one side of parallelogram is triple the other, perimeter 64, find longer side. 📏", "options": {"A": "12", "B": "16", "C": "20", "D": "24"}, "correct_answer": "D"},
    {"id": 56, "difficulty": "hard", "question": "Parallelogram with ∠A=120°, AB=10, AD=6. Distance between longer sides? 📏", "options": {"A": "3√3", "B": "4√3", "C": "5√3", "D": "6√3"}, "correct_answer": "C"},
    {"id": 57, "difficulty": "hard", "question": "Diagonals of quadrilateral intersect at right angles. Quadrilateral is rhombus if: 💎", "options": {"A": "Diagonals bisect", "B": "All sides equal", "C": "Diagonals equal", "D": "Opposite angles equal"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Parallelogram area=72, heights corresponding to sides 12 and 18 are? 📏", "options": {"A": "6 and 4", "B": "8 and 6", "C": "9 and 6", "D": "10 and 8"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "In rectangle, midpoint of longer side joined to opposite vertices divides rectangle into 3 equal area parts. Ratio of sides? ⬛", "options": {"A": "2:1", "B": "3:1", "C": "4:1", "D": "√3:1"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "ABCD is parallelogram. O is intersection of diagonals. If area(AOB)=10, area(ABCD)=? 📊", "options": {"A": "20", "B": "40", "C": "60", "D": "80"}, "correct_answer": "B"}
  ]
}