{
  "title": "Criteria for Similarity - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Two triangles are similar if their corresponding angles are equal. This is which criterion? 📐", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "If two triangles have their corresponding sides proportional, they are similar by which rule? ✨", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "AA"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "Which criterion states that if one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then triangles are similar? 🔍", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If two angles of one triangle are respectively equal to two angles of another triangle, the triangles are similar. This is called? 🧩", "options": {"A": "AA criterion", "B": "SSS criterion", "C": "SAS criterion", "D": "RHS criterion"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Which criterion is specifically for right-angled triangles similarity? 🔺", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "D"},
    {"id": 6, "difficulty": "easy", "question": "Two triangles with sides 3,4,5 and 6,8,10 are similar by which criterion? 📏", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "If ∠A = ∠D and ∠B = ∠E in ΔABC and ΔDEF, then the triangles are similar by? 🔄", "options": {"A": "AA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "For two triangles to be similar by SAS criterion, what must be proportional? ➗", "options": {"A": "All sides", "B": "Two sides and included angle", "C": "Two angles", "D": "Hypotenuse and one side"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Two right triangles are similar if the ratio of their corresponding sides is equal. True or False? 📐", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Only if acute angles equal"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "If in ΔABC and ΔPQR, AB/PQ = BC/QR = CA/RP, then they are similar by? 🧮", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "AA"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Which is NOT a criterion for similarity of triangles? ❌", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "SSA"}, "correct_answer": "D"},
    {"id": 12, "difficulty": "easy", "question": "If two triangles have all angles equal, they are similar by? ✨", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Two triangles with sides proportional are always similar. True or False? ⚖️", "options": {"A": "True", "B": "False", "C": "Only for right triangles", "D": "Only if angles known"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "For RHS similarity in right triangles, what must be proportional? 🔺", "options": {"A": "Hypotenuse and one side", "B": "Both legs", "C": "All three sides", "D": "Two angles"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "If ∠P = ∠X and PQ/XY = PR/XZ in ΔPQR and ΔXYZ, what criterion applies? 📏", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "All equilateral triangles are similar. Which criterion explains this? 🔵", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "AA"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Two isosceles triangles are always similar. True or False? 📐", "options": {"A": "True", "B": "False", "C": "Only if base angles equal", "D": "Only if vertex angles equal"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "If two triangles have two sides proportional and the non-included angle equal, are they similar? 🤔", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only if right-angled"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "Which criterion requires all three angles to be equal? 🎯", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "If two right triangles have an acute angle equal, they are similar by? 🔺", "options": {"A": "AA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 21, "difficulty": "medium", "question": "In ΔABC and ΔDEF, ∠A=∠D=50°, ∠B=∠E=60°, AB=4 cm, DE=6 cm, BC=6 cm, EF=9 cm. Are they similar? By which criterion? 📐", "options": {"A": "Yes, AA", "B": "Yes, SAS", "C": "Yes, SSS", "D": "No"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Two triangles have sides 5,12,13 and 10,24,26. Are they similar? If yes, by which rule? 🔍", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, RHS", "D": "No"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "In ΔPQR, PQ=8 cm, QR=10 cm, PR=12 cm. In ΔXYZ, XY=12 cm, YZ=15 cm, XZ=18 cm. Are they similar? ✨", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, AA", "D": "No"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If ΔABC ~ ΔDEF by SAS criterion, which of the following must be true? 🧩", "options": {"A": "∠A=∠D and AB/DE=AC/DF", "B": "AB/DE=BC/EF=CA/FD", "C": "∠A=∠D, ∠B=∠E", "D": "∠C=∠F and BC/EF=AC/DF"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Two right triangles have their legs proportional. Are they necessarily similar? 🔺", "options": {"A": "Yes", "B": "No", "C": "Only if hypotenuse also proportional", "D": "Only if one acute angle equal"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "In ΔABC, ∠A=40°, ∠B=60°. In ΔPQR, ∠P=40°, ∠Q=80°. Are the triangles similar? ⚖️", "options": {"A": "Yes", "B": "No", "C": "Cannot determine", "D": "Only if sides given"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "If two triangles are similar by AAA criterion, how many angles need to be equal? 🔢", "options": {"A": "2", "B": "3", "C": "1", "D": "All angles equal automatically if 2 are equal"}, "correct_answer": "D"},
    {"id": 28, "difficulty": "medium", "question": "For RHS similarity, the right angle must be between the proportional sides. True or False? 📏", "options": {"A": "True", "B": "False", "C": "Only for congruence", "D": "Not required"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "Two triangles with sides 7,24,25 and 14,48,50 are similar by which criterion? 🧮", "options": {"A": "SSS", "B": "RHS", "C": "SAS", "D": "AAA"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "If in two triangles, two angles of one are 30° and 70°, and two angles of the other are 80° and 30°, are they similar? 🌟", "options": {"A": "Yes", "B": "No", "C": "Cannot say", "D": "Only if sides proportional"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Which condition is sufficient to prove two triangles similar? ✅", "options": {"A": "Two sides proportional", "B": "Two angles equal", "C": "One angle equal", "D": "Perimeter equal"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "In ΔABC and ΔDEF, AB/DE = BC/EF and ∠B=∠E. Which criterion applies? 🔄", "options": {"A": "SAS", "B": "SSS", "C": "AA", "D": "RHS"}, "correct_answer": "A"},
    {"id": 33, "difficulty": "medium", "question": "Two triangles with angles 45°, 45°, 90° and 45°, 45°, 90° are similar by? 📐", "options": {"A": "AA", "B": "SSS", "C": "SAS", "D": "AAA"}, "correct_answer": "A"},
    {"id": 34, "difficulty": "medium", "question": "If two triangles have their corresponding sides in ratio 2:3, they are similar by? ➗", "options": {"A": "SSS", "B": "SAS", "C": "AA", "D": "RHS"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "For two polygons to be similar, both corresponding angles and sides must be proportional. True or False? 🔶", "options": {"A": "True", "B": "False", "C": "Only for triangles", "D": "Only for quadrilaterals"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If ΔABC ~ ΔPQR by SAS, and AB=6 cm, AC=8 cm, ∠A=50°, PQ=9 cm, what should be PR and ∠P? 📏", "options": {"A": "PR=12 cm, ∠P=50°", "B": "PR=8 cm, ∠P=50°", "C": "PR=12 cm, ∠P=40°", "D": "PR=9 cm, ∠P=50°"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "Two triangles with sides 8,15,17 and 16,30,34 are similar by? 🧩", "options": {"A": "SSS", "B": "RHS", "C": "SAS", "D": "Both A and B"}, "correct_answer": "A"},
    {"id": 38, "difficulty": "medium", "question": "If two triangles have two angles equal, the third angle is automatically equal. Why? 🤔", "options": {"A": "Angle sum property", "B": "Corresponding angles", "C": "Alternate angles", "D": "Vertically opposite"}, "correct_answer": "A"},
    {"id": 39, "difficulty": "medium", "question": "Which of the following pairs are always similar? ✨", "options": {"A": "Two squares", "B": "Two rectangles", "C": "Two rhombuses", "D": "Two trapeziums"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "In ΔABC, AB=5, BC=7, CA=9. In ΔDEF, DE=10, EF=14, FD=18. Are they similar? By which rule? 🔢", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, AA", "D": "No"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "hard", "question": "In ΔABC, D and E are on AB and AC such that DE∥BC. Which similarity criterion applies between ΔADE and ΔABC? 📐", "options": {"A": "AA", "B": "SSS", "C": "SAS", "D": "RHS"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "Two triangles have sides a,b,c and ka,kb,kc where k>0. Are they similar? By which criterion? 🔍", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, AAA", "D": "Yes, RHS"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "In ΔABC, ∠B=90°, BD⟂AC. Which triangles are similar? Name the criterion. 🔺", "options": {"A": "ΔADB~ΔBDC~ΔABC, AA", "B": "ΔADB~ΔABC, RHS", "C": "ΔBDC~ΔABC, SAS", "D": "ΔADB~ΔBDC, SSS"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "If two triangles are similar by AAA, are they necessarily similar by SSS? ⚖️", "options": {"A": "Yes", "B": "No", "C": "Only if sides given", "D": "Only if right-angled"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "In quadrilateral ABCD, diagonal AC divides it into two triangles. If AB/DC = BC/AD and ∠BAC=∠DCA, are ΔABC and ΔCDA similar? By which criterion? 🔄", "options": {"A": "Yes, SAS", "B": "Yes, SSS", "C": "Yes, AA", "D": "No"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "Two triangles have angles 30°, 60°, 90° and 30°, 60°, 90°. If the sides of the first are 1, √3, 2 and second are 2, 2√3, 4, by which criteria are they similar? 🧮", "options": {"A": "AA and SSS", "B": "SAS only", "C": "RHS only", "D": "AAA only"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "If two triangles have their corresponding altitudes proportional to corresponding sides, can we say they are similar? 🏔️", "options": {"A": "Yes, by SSS", "B": "Yes, by SAS", "C": "Yes, by AA", "D": "Not necessarily"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "hard", "question": "In ΔABC, ∠A=2∠B and ∠C=3∠B. In ΔPQR, ∠P=40°, ∠Q=20°, ∠R=120°. Are they similar? ✨", "options": {"A": "Yes", "B": "No", "C": "Cannot determine", "D": "Only if sides proportional"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "hard", "question": "Two triangles have sides (5,6,7) and (10,12,14). Are they similar? If yes, by which criteria? 📏", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, AAA", "D": "No"}, "correct_answer": "A"},
    {"id": 50, "difficulty": "hard", "question": "If ΔABC ~ ΔDEF by SAS, with AB/DE = AC/DF and ∠A=∠D, and BC=12 cm, EF=18 cm, find the ratio of their areas. 📐", "options": {"A": "4:9", "B": "2:3", "C": "12:18", "D": "√2:√3"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "In two triangles, two sides of one are proportional to two sides of the other, and the median to the third side is proportional. Are they similar? ⚕️", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only if angles equal"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "hard", "question": "Two triangles have perimeters 30 cm and 45 cm. If they are similar, what is the ratio of their corresponding sides? 📊", "options": {"A": "2:3", "B": "3:2", "C": "30:45", "D": "√30:√45"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "If in ΔABC and ΔPQR, ∠A=∠P, ∠B=∠Q, and AB/PQ = BC/QR, are they similar? By which criterion? 🔍", "options": {"A": "Yes, AA", "B": "Yes, SAS", "C": "Yes, SSS", "D": "Not necessarily"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "Two right triangles have one acute angle equal. Which similarity criterion is satisfied? 🔺", "options": {"A": "AA", "B": "RHS", "C": "SAS", "D": "SSS"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "In ΔABC, DE∥BC with D on AB, E on AC. If AD=3 cm, DB=2 cm, AE=4.5 cm, find EC. Which similarity criterion is used? 📏", "options": {"A": "3 cm, AA", "B": "3 cm, SSS", "C": "2.5 cm, SAS", "D": "3 cm, BPT"}, "correct_answer": "A"},
    {"id": 56, "difficulty": "hard", "question": "Two triangles have their corresponding medians in ratio 3:4. If they are similar, what is the ratio of their sides? ⚕️", "options": {"A": "3:4", "B": "9:16", "C": "√3:2", "D": "4:3"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "If two triangles are similar by SSS, are their corresponding angles equal? ✅", "options": {"A": "Yes", "B": "No", "C": "Only if right-angled", "D": "Only if isosceles"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "In ΔABC and ΔDEF, AB/DE = BC/EF = CA/FD = 2/3. Also, ∠A=50°, ∠D=50°. Which criteria prove similarity? 🧩", "options": {"A": "SSS and SAS", "B": "AAA only", "C": "SAS only", "D": "SSS only"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "Two triangles have sides (8,10,12) and (12,15,18). Are they similar? If yes, by which criteria? 🔢", "options": {"A": "Yes, SSS", "B": "Yes, SAS", "C": "Yes, AA", "D": "No"}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If two triangles are similar, their corresponding angle bisectors are proportional to their sides. Which similarity criterion indirectly implies this? 🔄", "options": {"A": "AAA", "B": "SSS", "C": "SAS", "D": "All"}, "correct_answer": "D"}
  ]
}