{
  "title": "Angle Properties of Circles - Grade 10 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "If ∠AOB is at the center of a circle 🎯 and ∠ACB is on the circumference subtended by the same arc AB, what is the relationship? 🤔", "options": {"A": "∠AOB = ∠ACB", "B": "∠AOB = 2∠ACB", "C": "∠ACB = 2∠AOB", "D": "∠AOB + ∠ACB = 180°"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "An angle in a semicircle is always a ______ angle. 📐", "options": {"A": "acute", "B": "obtuse", "C": "right", "D": "straight"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "Angles in the same segment of a circle are ______. 🔄", "options": {"A": "complementary", "B": "supplementary", "C": "equal", "D": "double"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If ∠ACB = 40° on the circumference, what is ∠AOB at the center? 🧮", "options": {"A": "20°", "B": "40°", "C": "80°", "D": "160°"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "The opposite angles of a cyclic quadrilateral are ______. ⭕", "options": {"A": "equal", "B": "supplementary", "C": "complementary", "D": "right angles"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If O is the center and ∠ABC = 50°, what is ∠AOC? 🤓", "options": {"A": "25°", "B": "50°", "C": "100°", "D": "200°"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "An angle at the center is ______ the angle at the circumference subtended by the same arc. ↔️", "options": {"A": "half of", "B": "equal to", "C": "double", "D": "triple"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "In a circle, the angle in a major segment is ______. 📏", "options": {"A": "acute", "B": "obtuse", "C": "right", "D": "straight"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "If ∠PQR = 110° in a cyclic quadrilateral PQRS, what is ∠PSR? 🔄", "options": {"A": "55°", "B": "70°", "C": "110°", "D": "180°"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "Angle formed in a minor segment is ______. 🔽", "options": {"A": "acute", "B": "obtuse", "C": "right", "D": "reflex"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "If arc AB subtends 60° at the center, what does it subtend at the circumference? 🧐", "options": {"A": "15°", "B": "30°", "C": "60°", "D": "120°"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "A cyclic quadrilateral ABCD has ∠A = 70°. What is ∠C? ⭕", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "180°"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "In a circle with center O, ∠OAB = 40°. What is ∠AOB if OA and OB are radii? 📐", "options": {"A": "40°", "B": "80°", "C": "100°", "D": "140°"}, "correct_answer": "C"},
    {"id": 14, "difficulty": "easy", "question": "Which angle is 90° in a circle? 🎯", "options": {"A": "Angle in a major segment", "B": "Angle in a semicircle", "C": "Angle at the center", "D": "Angle in same segment"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "If two angles subtend the same chord on the same side, they are ______. 🔗", "options": {"A": "different", "B": "equal", "C": "supplementary", "D": "complementary"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "The exterior angle of a cyclic quadrilateral equals the ______ interior opposite angle. ↔️", "options": {"A": "adjacent", "B": "corresponding", "C": "alternate", "D": "opposite"}, "correct_answer": "D"},
    {"id": 17, "difficulty": "easy", "question": "In a circle, if ∠XYZ = 25°, what is the reflex ∠XOZ? (O is center) 🤔", "options": {"A": "25°", "B": "50°", "C": "130°", "D": "310°"}, "correct_answer": "D"},
    {"id": 18, "difficulty": "easy", "question": "A chord makes equal angles at ______ of the circle. 🎵", "options": {"A": "center", "B": "circumference on same side", "C": "circumference on opposite sides", "D": "outside"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If ∠APB = 35° where A,B are on circle and P is on circumference, what is ∠AOB? (O is center) 📍", "options": {"A": "17.5°", "B": "35°", "C": "70°", "D": "105°"}, "correct_answer": "C"},
    {"id": 20, "difficulty": "easy", "question": "The sum of either pair of opposite angles in a cyclic quadrilateral is ______. ➕", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "In a circle, chord AB subtends 50° at center O. What angle does it subtend at point C on the major arc? 🧮", "options": {"A": "25°", "B": "50°", "C": "100°", "D": "130°"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "ABCD is a cyclic quadrilateral with ∠A = 3x and ∠C = 2x + 30. Find x. ⚖️", "options": {"A": "20°", "B": "30°", "C": "40°", "D": "50°"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "In a circle, O is center. Chord PQ makes ∠POQ = 120°. What is ∠PRQ where R is on minor arc? 📐", "options": {"A": "30°", "B": "60°", "C": "120°", "D": "240°"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "Two chords AB and AC have ∠BAC = 40° inside the circle. If O is center, find ∠BOC. 🔍", "options": {"A": "40°", "B": "80°", "C": "100°", "D": "140°"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Points A,B,C,D lie on a circle. ∠ABC = 70°, ∠BCD = 90°. Find ∠ADC. 🤔", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "160°"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "The angle between a chord and tangent at one endpoint equals angle in the ______ segment. 🔗", "options": {"A": "same", "B": "opposite", "C": "alternate", "D": "major"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "In cyclic quadrilateral PQRS, ∠P:∠Q:∠R = 4:5:6. Find ∠S. 📊", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Chord AB = chord AC. If ∠BAC = 50°, find angle subtended by BC at center. ⚙️", "options": {"A": "50°", "B": "80°", "C": "100°", "D": "160°"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "O is center of circle. ∠OAT = 90° where AT is tangent. If ∠AOB = 110°, find ∠ABT. 🎯", "options": {"A": "35°", "B": "55°", "C": "70°", "D": "110°"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "In a circle, two chords AB & CD intersect at X inside. If ∠AXC = 100° and arc AD = 60°, find arc BC. 🧩", "options": {"A": "40°", "B": "100°", "C": "140°", "D": "160°"}, "correct_answer": "C"},
    {"id": 31, "difficulty": "medium", "question": "ABCD is cyclic. ∠A = 75°, ∠B = 105°. Find ∠C and ∠D difference. 🔄", "options": {"A": "0°", "B": "30°", "C": "60°", "D": "90°"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "Angle between radius and tangent at point of contact is ______. 📐", "options": {"A": "acute", "B": "obtuse", "C": "right", "D": "straight"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "From point P outside circle, two tangents PA & PB are drawn. If ∠APB = 60°, find ∠AOB. (O is center) 🤓", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "180°"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "Chords PQ & RS intersect at T inside circle. If ∠PTS = 85° and arc PR = 40°, find arc QS. 🎵", "options": {"A": "45°", "B": "85°", "C": "130°", "D": "170°"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "In a circle, ∠ABC = 55° where A,B,C are on circle. Find reflex ∠AOC if O is center. 🔄", "options": {"A": "55°", "B": "110°", "C": "250°", "D": "290°"}, "correct_answer": "D"},
    {"id": 36, "difficulty": "medium", "question": "The angle in a semicircle is 90°. What is angle in a quadrant? 📏", "options": {"A": "45°", "B": "90°", "C": "135°", "D": "180°"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "If a cyclic quadrilateral has one angle 80°, what is the opposite angle? ⚖️", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "180°"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "Angle subtended by diameter at circumference is ______. 🌕", "options": {"A": "acute", "B": "right", "C": "obtuse", "D": "straight"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "In circle, chord MN subtends 45° at center. What angle does it subtend at point on minor arc? 🔍", "options": {"A": "22.5°", "B": "45°", "C": "67.5°", "D": "135°"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "O is center. Chord AB = radius. What is ∠AOB? 📏", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 41, "difficulty": "hard", "question": "In a circle, chords AB & CD intersect at X. AX = 6cm, XB = 4cm, CX = 3cm. Find DX using intersecting chords theorem. 📏", "options": {"A": "4.5 cm", "B": "6 cm", "C": "8 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "hard", "question": "Two circles intersect at P & Q. Lines through P meet circles at A & B. Lines through Q meet circles at C & D. Prove AC || BD. Which theorem applies? 🤯", "options": {"A": "Alternate segment theorem", "B": "Intersecting chords theorem", "C": "Cyclic quadrilateral property", "D": "Angles in same segment"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "From point P outside circle, secant PAB and tangent PT drawn. If PA = 4cm, AB = 5cm, find PT length. 🧮", "options": {"A": "3 cm", "B": "5 cm", "C": "6 cm", "D": "9 cm"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "In cyclic quadrilateral ABCD, AB = AD and CB = CD. If ∠BAD = 100°, find ∠BCD. 🔄", "options": {"A": "60°", "B": "80°", "C": "100°", "D": "120°"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "Chords PQ & RS intersect at T. PT = 5, TQ = 3, RT = 4. Find ST using chord theorem. 📐", "options": {"A": "3.75", "B": "4.5", "C": "6", "D": "7.5"}, "correct_answer": "A"},
    {"id": 46, "difficulty": "hard", "question": "Two tangents PA & PB from point P to circle with center O. If ∠APB = 50°, find ∠AOB. 🎯", "options": {"A": "115°", "B": "130°", "C": "150°", "D": "180°"}, "correct_answer": "B"},
    {"id": 47, "difficulty": "hard", "question": "In circle, diameter AB. Chord CD is perpendicular to AB at point P. If AP = 2cm, PB = 8cm, find CP. 📏", "options": {"A": "2 cm", "B": "3 cm", "C": "4 cm", "D": "5 cm"}, "correct_answer": "C"},
    {"id": 48, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD has ∠A = 7x, ∠B = 5x, ∠C = 4x. Find ∠D. ⚙️", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Two circles intersect at X & Y. Line through X cuts circles at A & B. Line through Y cuts circles at C & D. If ∠XBY = 40°, find ∠XAY. 🔍", "options": {"A": "40°", "B": "50°", "C": "80°", "D": "140°"}, "correct_answer": "D"},
    {"id": 50, "difficulty": "hard", "question": "From point P, tangent PT = 10cm, secant PAB with PA = 8cm. Find length of chord AB. 📊", "options": {"A": "2.5 cm", "B": "3 cm", "C": "4.5 cm", "D": "6 cm"}, "correct_answer": "C"},
    {"id": 51, "difficulty": "hard", "question": "In circle, O is center. Chord AB = radius. Another chord BC = radius. Find ∠ABC. 🤔", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "75°"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "hard", "question": "Cyclic quadrilateral has sides subtending angles at center: 80°, 100°, 120°, 60°. Find smallest interior angle. ⚖️", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Two circles touch internally at P. Chord PQ of larger circle touches smaller circle at R. Prove PR = RQ. Which theorem? 🎓", "options": {"A": "Tangent-secant theorem", "B": "Alternate segment theorem", "C": "Intersecting chords theorem", "D": "Pythagoras theorem"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "In circle, chords AB & CD intersect at X. AX = 8, BX = 6, CX:DX = 4:3. Find CX. 🧩", "options": {"A": "4", "B": "6", "C": "8", "D": "10"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "hard", "question": "From external point P, two tangents to circle with center O. If ∠OPA = 30° and radius = 5cm, find PA length. 📏", "options": {"A": "5 cm", "B": "5√3 cm", "C": "10 cm", "D": "10√3 cm"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "In cyclic quadrilateral ABCD, diagonals intersect at E. If ∠ADB = 40°, ∠CBD = 30°, find ∠AEB. 🔄", "options": {"A": "60°", "B": "70°", "C": "80°", "D": "110°"}, "correct_answer": "D"},
    {"id": 57, "difficulty": "hard", "question": "Two chords intersect at right angles inside circle. Segments of one chord are 4cm & 9cm. Find radius if distance from center to chord is 3cm. 🌐", "options": {"A": "5 cm", "B": "6 cm", "C": "7 cm", "D": "8 cm"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "hard", "question": "Three points A,B,C on circle with ∠ABC = 50°. Tangent at C meets AB extended at D. Find ∠BCD. 🔍", "options": {"A": "30°", "B": "40°", "C": "50°", "D": "60°"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "hard", "question": "In circle, diameter AB. Point C on circle such that AC = BC. Find ∠CAB. 📐", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "hard", "question": "Two circles intersect at P & Q. Lines through P meet circles at A & B. Lines through Q meet circles at C & D. If ∠PAC = 70°, find ∠PBD. 🔗", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "140°"}, "correct_answer": "C"}
  ]
}