{
"title": "Cyclic Properties (Circles) - Grade 10 ICSE",
"total_questions": 60,
"questions": [
{"id": 1, "difficulty": "easy", "question": "A quadrilateral whose all vertices lie on a circle is called ______. ⭕", "options": {"A": "cyclic quadrilateral", "B": "regular quadrilateral", "C": "tangential quadrilateral", "D": "square"}, "correct_answer": "A"},
{"id": 2, "difficulty": "easy", "question": "Opposite angles of a cyclic quadrilateral are ______. 🔄", "options": {"A": "equal", "B": "complementary", "C": "supplementary", "D": "right angles"}, "correct_answer": "C"},
{"id": 3, "difficulty": "easy", "question": "In a cyclic quadrilateral, the exterior angle equals the ______ interior opposite angle. ↔️", "options": {"A": "adjacent", "B": "corresponding", "C": "alternate", "D": "opposite"}, "correct_answer": "D"},
{"id": 4, "difficulty": "easy", "question": "If ∠A = 70° in cyclic quadrilateral ABCD, then ∠C = ______. 🧮", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "180°"}, "correct_answer": "C"},
{"id": 5, "difficulty": "easy", "question": "The sum of opposite angles in a cyclic quadrilateral is ______. ➕", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "B"},
{"id": 6, "difficulty": "easy", "question": "If ∠P = 80° in cyclic quadrilateral PQRS, then ∠R = ______. 📐", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "180°"}, "correct_answer": "B"},
{"id": 7, "difficulty": "easy", "question": "A cyclic parallelogram must be a ______. 🔲", "options": {"A": "square", "B": "rhombus", "C": "rectangle", "D": "trapezium"}, "correct_answer": "C"},
{"id": 8, "difficulty": "easy", "question": "If one angle of a cyclic quadrilateral is 120°, its opposite angle is ______. ⚖️", "options": {"A": "60°", "B": "120°", "C": "150°", "D": "180°"}, "correct_answer": "A"},
{"id": 9, "difficulty": "easy", "question": "A quadrilateral ABCD is cyclic if ∠A + ∠C = ______. 🤔", "options": {"A": "90°", "B": "180°", "C": "270°", "D": "360°"}, "correct_answer": "B"},
{"id": 10, "difficulty": "easy", "question": "In cyclic quadrilateral, exterior angle at A equals interior angle at ______. 🔍", "options": {"A": "B", "B": "C", "C": "D", "D": "opposite vertex"}, "correct_answer": "B"},
{"id": 11, "difficulty": "easy", "question": "All rectangles are ______. 📦", "options": {"A": "cyclic", "B": "not cyclic", "C": "sometimes cyclic", "D": "never cyclic"}, "correct_answer": "A"},
{"id": 12, "difficulty": "easy", "question": "If ∠B = 95° in cyclic quadrilateral ABCD, then ∠D = ______. 🎯", "options": {"A": "85°", "B": "95°", "C": "105°", "D": "175°"}, "correct_answer": "A"},
{"id": 13, "difficulty": "easy", "question": "A cyclic trapezium is always ______. 📏", "options": {"A": "scalene", "B": "right-angled", "C": "isosceles", "D": "obtuse-angled"}, "correct_answer": "C"},
{"id": 14, "difficulty": "easy", "question": "The four vertices of a square lie on a circle. True or false? ✅❌", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Never"}, "correct_answer": "A"},
{"id": 15, "difficulty": "easy", "question": "In a cyclic quadrilateral, if three angles are 70°, 110°, 80°, the fourth angle is ______. 🔢", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "110°"}, "correct_answer": "C"},
{"id": 16, "difficulty": "easy", "question": "A kite is always cyclic. True or false? 🪁", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Always"}, "correct_answer": "B"},
{"id": 17, "difficulty": "easy", "question": "Exterior angle ∠CBE of cyclic quadrilateral ABCD equals interior angle at ______. ↔️", "options": {"A": "A", "B": "B", "C": "C", "D": "D"}, "correct_answer": "D"},
{"id": 18, "difficulty": "easy", "question": "If a quadrilateral is cyclic, its perpendicular bisectors of sides are ______. ⚡", "options": {"A": "parallel", "B": "perpendicular", "C": "concurrent", "D": "equal"}, "correct_answer": "C"},
{"id": 19, "difficulty": "easy", "question": "A rhombus is cyclic only if it is a ______. 🔶", "options": {"A": "square", "B": "rectangle", "C": "trapezium", "D": "parallelogram"}, "correct_answer": "A"},
{"id": 20, "difficulty": "easy", "question": "In cyclic quadrilateral, angles in the same segment are ______. 🔗", "options": {"A": "equal", "B": "supplementary", "C": "complementary", "D": "right angles"}, "correct_answer": "A"},
{"id": 21, "difficulty": "medium", "question": "In cyclic quadrilateral ABCD, ∠A = 2x+10, ∠C = 3x-20. Find x. ⚙️", "options": {"A": "30°", "B": "38°", "C": "40°", "D": "50°"}, "correct_answer": "B"},
{"id": 22, "difficulty": "medium", "question": "ABCD is cyclic. AB∥DC. If ∠A = 65°, find ∠C. 🔄", "options": {"A": "65°", "B": "105°", "C": "115°", "D": "125°"}, "correct_answer": "C"},
{"id": 23, "difficulty": "medium", "question": "If cyclic quadrilateral has angles (2x)°, (3x)°, (4x)°, (5x)°, find largest angle. 📊", "options": {"A": "90°", "B": "120°", "C": "150°", "D": "180°"}, "correct_answer": "C"},
{"id": 24, "difficulty": "medium", "question": "Two circles intersect at P & Q. Lines through P meet circles at A & B. Lines through Q meet circles at C & D. Then quadrilateral ACBD is ______. 🤔", "options": {"A": "cyclic", "B": "parallelogram", "C": "trapezium", "D": "square"}, "correct_answer": "A"},
{"id": 25, "difficulty": "medium", "question": "In cyclic quadrilateral, ratio of angles is 2:3:4:3. Find smallest angle. 🧮", "options": {"A": "40°", "B": "60°", "C": "80°", "D": "90°"}, "correct_answer": "B"},
{"id": 26, "difficulty": "medium", "question": "ABCD is cyclic. ∠A = 70°, ∠B = 120°. Find difference ∠D - ∠C. ➖", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "50°"}, "correct_answer": "D"},
{"id": 27, "difficulty": "medium", "question": "In circle, chords AB & CD intersect at X. Then quadrilateral ACBD is cyclic if ______. 🔗", "options": {"A": "AX·XB = CX·XD", "B": "AX = XB", "C": "∠AXC = 90°", "D": "AB = CD"}, "correct_answer": "A"},
{"id": 28, "difficulty": "medium", "question": "Cyclic quadrilateral PQRS has ∠P:∠Q = 3:2, ∠Q:∠R = 2:1. Find ∠S. 🔄", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "B"},
{"id": 29, "difficulty": "medium", "question": "Exterior angle at A of cyclic quadrilateral ABCD is 110°. Find ∠C. 📐", "options": {"A": "70°", "B": "90°", "C": "110°", "D": "180°"}, "correct_answer": "C"},
{"id": 30, "difficulty": "medium", "question": "A cyclic trapezium ABCD with AB∥DC has ∠A = 100°. Find ∠C. 📏", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "180°"}, "correct_answer": "A"},
{"id": 31, "difficulty": "medium", "question": "In cyclic quadrilateral, one diagonal is diameter. If ∠ABD = 40°, find ∠ADB. 🎯", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
{"id": 32, "difficulty": "medium", "question": "If a parallelogram is cyclic, it must be a ______. 🔲", "options": {"A": "square", "B": "rhombus", "C": "rectangle", "D": "kite"}, "correct_answer": "C"},
{"id": 33, "difficulty": "medium", "question": "Points A,B,C,D lie on circle. ∠ABC = 130°, ∠ADC = 50°. Is ABCD cyclic? 🤨", "options": {"A": "Yes", "B": "No", "C": "Cannot say", "D": "Only if AB=CD"}, "correct_answer": "A"},
{"id": 34, "difficulty": "medium", "question": "In cyclic quadrilateral, exterior angle at vertex B is 95°. Find interior angle at D. 🔍", "options": {"A": "85°", "B": "95°", "C": "105°", "D": "180°"}, "correct_answer": "B"},
{"id": 35, "difficulty": "medium", "question": "Two chords AB & CD intersect at X. If AX·XB = CX·XD, then A,B,C,D are ______. ⭕", "options": {"A": "collinear", "B": "concyclic", "C": "vertices of square", "D": "on two circles"}, "correct_answer": "B"},
{"id": 36, "difficulty": "medium", "question": "In cyclic quadrilateral, sum of opposite angles is 180°. What is sum of all angles? ➕", "options": {"A": "180°", "B": "270°", "C": "360°", "D": "540°"}, "correct_answer": "C"},
{"id": 37, "difficulty": "medium", "question": "If ∠A = 85° and ∠C = 105° in quadrilateral ABCD, can it be cyclic? 🤔", "options": {"A": "Yes", "B": "No", "C": "Maybe", "D": "Only if ∠B=∠D"}, "correct_answer": "B"},
{"id": 38, "difficulty": "medium", "question": "Cyclic quadrilateral ABCD has AB = AD, CB = CD. If ∠BAD = 80°, find ∠BCD. 🔄", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "B"},
{"id": 39, "difficulty": "medium", "question": "If three vertices of quadrilateral lie on circle and opposite angles supplementary, the fourth vertex ______. 🎯", "options": {"A": "lies inside", "B": "lies outside", "C": "lies on circle", "D": "cannot say"}, "correct_answer": "C"},
{"id": 40, "difficulty": "medium", "question": "In cyclic quadrilateral, diagonals intersect at right angles. One angle is 75°. Find opposite angle. 📐", "options": {"A": "75°", "B": "105°", "C": "115°", "D": "165°"}, "correct_answer": "B"},
{"id": 41, "difficulty": "hard", "question": "Two circles intersect at P & Q. Through P, lines APB & CPD are drawn meeting circles at A,B,C,D. Prove ACBD is cyclic. Which theorem? 🧠", "options": {"A": "Angles in same segment", "B": "Exterior angle equals interior opposite", "C": "Intersecting chords theorem", "D": "Pythagoras theorem"}, "correct_answer": "B"},
{"id": 42, "difficulty": "hard", "question": "In cyclic quadrilateral ABCD, diagonals intersect at E. ∠ABD = 30°, ∠DBC = 40°. Find ∠AEB. 🔍", "options": {"A": "60°", "B": "70°", "C": "80°", "D": "110°"}, "correct_answer": "D"},
{"id": 43, "difficulty": "hard", "question": "A circle is inscribed in quadrilateral ABCD touching sides at P,Q,R,S. Then ABCD is cyclic if ______. 🔄", "options": {"A": "AB+CD = BC+AD", "B": "AP = AS", "C": "∠A+∠C = 180°", "D": "PQRS is square"}, "correct_answer": "A"},
{"id": 44, "difficulty": "hard", "question": "Two chords AB & CD intersect at X. AX = 6, XB = 4, CX = 3. Find DX if points are concyclic. 📏", "options": {"A": "2", "B": "4.5", "C": "6", "D": "8"}, "correct_answer": "D"},
{"id": 45, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD has AB = 6, BC = 7, CD = 8, DA = 9. Find product of diagonals if angle between them is 60°. 🧮", "options": {"A": "42", "B": "42√3", "C": "84", "D": "84√3"}, "correct_answer": "C"},
{"id": 46, "difficulty": "hard", "question": "From point P outside circle, two secants PAB & PCD drawn. If PA=4, AB=6, PC=3, find CD using cyclic property. 📐", "options": {"A": "4", "B": "5", "C": "6", "D": "8"}, "correct_answer": "B"},
{"id": 47, "difficulty": "hard", "question": "Two circles intersect at X & Y. Through X line AXB drawn. Through Y line CYD drawn. If ∠XBY = 35°, find ∠XAY. 🤯", "options": {"A": "35°", "B": "55°", "C": "145°", "D": "215°"}, "correct_answer": "C"},
{"id": 48, "difficulty": "hard", "question": "In cyclic quadrilateral, sides subtend angles at center: 70°, 110°, 130°, 50°. Find smallest interior angle. ⚖️", "options": {"A": "25°", "B": "35°", "C": "55°", "D": "65°"}, "correct_answer": "D"},
{"id": 49, "difficulty": "hard", "question": "ABCD is cyclic. Diagonals AC & BD intersect at E. If ∠ABD = 25°, ∠DBC = 35°, ∠ACB = 40°, find ∠ADB. 🔄", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "80°"}, "correct_answer": "D"},
{"id": 50, "difficulty": "hard", "question": "Two circles intersect at P & Q. Tangents at P to circles meet circles again at R & S. Then quadrilateral PQRS is ______. 🔗", "options": {"A": "cyclic", "B": "parallelogram", "C": "trapezium", "D": "kite"}, "correct_answer": "A"},
{"id": 51, "difficulty": "hard", "question": "In cyclic quadrilateral ABCD, ∠A = 60°, ∠B = 90°, AB = 5, BC = 12. Find diagonal AC using Ptolemy's theorem hint. 📏", "options": {"A": "10", "B": "13", "C": "15", "D": "17"}, "correct_answer": "B"},
{"id": 52, "difficulty": "hard", "question": "Cyclic quadrilateral has perpendicular diagonals. If sides are 5,6,7,8, find product of diagonals. 🧩", "options": {"A": "√210", "B": "√410", "C": "√610", "D": "√810"}, "correct_answer": "B"},
{"id": 53, "difficulty": "hard", "question": "Two chords AB & CD intersect at X. A circle through A,C,X cuts BD at Y. Prove X,Y,C,A concyclic uses which theorem? 🎓", "options": {"A": "Alternate segment", "B": "Intersecting chords", "C": "Cyclic quadrilateral exterior angle", "D": "Tangent properties"}, "correct_answer": "C"},
{"id": 54, "difficulty": "hard", "question": "Cyclic quadrilateral ABCD has ∠A = 70°, AB = 8, BC = 6, CD = 7, DA = 9. Find approximate area. 📐", "options": {"A": "30", "B": "35", "C": "40", "D": "45"}, "correct_answer": "D"},
{"id": 55, "difficulty": "hard", "question": "From point P outside circle, two tangents PT & PS, secant PAB. If PA=4, AB=5, PT=6, find PS. 🔄", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "B"},
{"id": 56, "difficulty": "hard", "question": "In cyclic quadrilateral, if sum of opposite sides equal, then ______. ↔️", "options": {"A": "diagonals equal", "B": "it's trapezium", "C": "circle can be inscribed", "D": "it's rectangle"}, "correct_answer": "C"},
{"id": 57, "difficulty": "hard", "question": "Two circles intersect at P & Q. Through P diameters PA & PB drawn. Then A,Q,B are ______. 🔗", "options": {"A": "collinear", "B": "vertices of cyclic quad", "C": "equidistant", "D": "on perpendicular lines"}, "correct_answer": "A"},
{"id": 58, "difficulty": "hard", "question": "In cyclic quadrilateral ABCD, ∠A = 2x, ∠B = 3x, ∠C = 4x, ∠D = 5x. Find x and state if possible. 🤔", "options": {"A": "x=20°, possible", "B": "x=25°, possible", "C": "x=25°, impossible", "D": "x=30°, impossible"}, "correct_answer": "C"},
{"id": 59, "difficulty": "hard", "question": "If quadrilateral has perpendicular diagonals and midpoints of sides concyclic, then quadrilateral is ______. ⭕", "options": {"A": "cyclic", "B": "tangential", "C": "parallelogram", "D": "isosceles trapezium"}, "correct_answer": "A"},
{"id": 60, "difficulty": "hard", "question": "Two chords intersect at X inside circle. Through X line perpendicular to one chord meets circle at Y,Z. Then XY = XZ if ______. 📏", "options": {"A": "chords equal", "B": "X is center", "C": "quadrilateral cyclic", "D": "chords perpendicular"}, "correct_answer": "C"}
]
}