{
  "title": "Vertically Opposite Angles - Grade 9 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "What is the measure of angle x if its vertically opposite angle is 45°?", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "If two lines intersect and one angle is 75°, what is its vertically opposite angle?", "options": {"A": "75°", "B": "105°", "C": "15°", "D": "90°"}, "correct_answer": "A"},
    {"id": 3, "difficulty": "easy", "question": "Vertically opposite angles are always:", "options": {"A": "Complementary", "B": "Supplementary", "C": "Equal", "D": "Unequal"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "If ∠A = 60° and is vertically opposite to ∠B, what is ∠B?", "options": {"A": "30°", "B": "120°", "C": "60°", "D": "90°"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "Two lines intersect forming four angles. If one angle is 40°, how many angles are 40°?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "Vertically opposite angles are formed by:", "options": {"A": "Parallel lines", "B": "Intersecting lines", "C": "Perpendicular lines", "D": "Curved lines"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "If ∠1 and ∠2 are vertically opposite and ∠1 = 85°, then ∠2 = ?", "options": {"A": "85°", "B": "95°", "C": "5°", "D": "175°"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "Which pair of angles are always equal when two lines intersect?", "options": {"A": "Adjacent angles", "B": "Linear pair", "C": "Vertically opposite", "D": "Corresponding angles"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "If x and y are vertically opposite angles, then:", "options": {"A": "x + y = 90°", "B": "x + y = 180°", "C": "x = y", "D": "x - y = 90°"}, "correct_answer": "C"},
    {"id": 10, "difficulty": "easy", "question": "In the figure, if ∠AOC = 50°, what is ∠BOD?", "options": {"A": "50°", "B": "130°", "C": "40°", "D": "140°"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "Two lines intersect. If one vertically opposite pair measures 70° each, what is the measure of the other pair?", "options": {"A": "70°", "B": "110°", "C": "20°", "D": "180°"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "Vertically opposite angles are equal because:", "options": {"A": "They are adjacent", "B": "They form linear pairs", "C": "They share a common vertex and are opposite", "D": "They are complementary"}, "correct_answer": "C"},
    {"id": 13, "difficulty": "easy", "question": "If two intersecting lines form an angle of 35°, what is its vertically opposite angle?", "options": {"A": "145°", "B": "35°", "C": "55°", "D": "125°"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "Which of these statements is true about vertically opposite angles?", "options": {"A": "They add up to 90°", "B": "They are always acute", "C": "They are equal", "D": "They are supplementary"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "In the diagram, if ∠1 = 120°, then ∠3 (vertically opposite to ∠1) is:", "options": {"A": "60°", "B": "120°", "C": "30°", "D": "240°"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "Two lines intersect at point O. If ∠AOB = 80°, then ∠COD (vertically opposite) is:", "options": {"A": "100°", "B": "80°", "C": "10°", "D": "280°"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "If the measure of a vertically opposite angle is 0°, what can you say about the lines?", "options": {"A": "They are parallel", "B": "They coincide", "C": "They are perpendicular", "D": "They are skew lines"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "When two lines intersect, how many pairs of vertically opposite angles are formed?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If ∠P and ∠Q are vertically opposite and ∠P = 90°, what type of lines are these?", "options": {"A": "Parallel", "B": "Coinciding", "C": "Perpendicular", "D": "Intersecting at acute angle"}, "correct_answer": "D"},
    {"id": 20, "difficulty": "easy", "question": "Two lines intersect. One angle is 25°. What is the angle vertically opposite to it?", "options": {"A": "155°", "B": "25°", "C": "65°", "D": "115°"}, "correct_answer": "B"},

    {"id": 21, "difficulty": "medium", "question": "Two lines intersect. If one angle is 2x + 10° and its vertically opposite angle is 50°, find x.", "options": {"A": "10", "B": "15", "C": "20", "D": "25"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "In the figure, lines AB and CD intersect at O. If ∠AOC = (3x + 20)° and ∠BOD = (2x + 40)°, find x.", "options": {"A": "10", "B": "20", "C": "30", "D": "40"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Two lines intersect forming four angles. If one angle is 20° less than its vertically opposite angle, find the angles.", "options": {"A": "80° and 100°", "B": "70° and 110°", "C": "90° and 90°", "D": "60° and 120°"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "If two lines intersect and one pair of vertically opposite angles are in ratio 2:3, find all four angles.", "options": {"A": "60°, 120°, 60°, 120°", "B": "72°, 108°, 72°, 108°", "C": "80°, 100°, 80°, 100°", "D": "90°, 90°, 90°, 90°"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "In the figure, if ∠AOD = 130° and ∠BOC = 3x + 10°, find x.", "options": {"A": "20", "B": "30", "C": "40", "D": "50"}, "correct_answer": "C"},
    {"id": 26, "difficulty": "medium", "question": "Two lines intersect. The sum of two adjacent angles is 150°. Find all four angles.", "options": {"A": "75°, 105°, 75°, 105°", "B": "60°, 120°, 60°, 120°", "C": "30°, 150°, 30°, 150°", "D": "90°, 90°, 90°, 90°"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "If ∠A and ∠B are vertically opposite angles and ∠A = 5x - 20°, ∠B = 3x + 40°, find the measure of ∠A.", "options": {"A": "100°", "B": "120°", "C": "140°", "D": "160°"}, "correct_answer": "D"},
    {"id": 28, "difficulty": "medium", "question": "In the figure, lines PQ and RS intersect at T. If ∠PTR = 5y° and ∠QTS = (2y + 60)°, find y.", "options": {"A": "10", "B": "15", "C": "20", "D": "25"}, "correct_answer": "C"},
    {"id": 29, "difficulty": "medium", "question": "Two lines intersect. One angle is three times its vertically opposite angle. Find the angles.", "options": {"A": "45° and 135°", "B": "60° and 120°", "C": "90° and 90°", "D": "30° and 150°"}, "correct_answer": "A"},
    {"id": 30, "difficulty": "medium", "question": "If the difference between two vertically opposite angles is 40°, find the angles.", "options": {"A": "40° and 140°", "B": "70° and 110°", "C": "80° and 120°", "D": "They can't have a difference"}, "correct_answer": "D"},
    {"id": 31, "difficulty": "medium", "question": "Lines AB and CD intersect at O. If ∠AOC + ∠BOD = 140°, find ∠AOC.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "In the figure, if ∠1 = (4x - 30)° and ∠3 = (2x + 50)° are vertically opposite, find x.", "options": {"A": "20", "B": "30", "C": "40", "D": "50"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "Two lines intersect. One pair of vertically opposite angles are (x + 30)° and (3x - 50)°. Find x.", "options": {"A": "20", "B": "30", "C": "40", "D": "50"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "If ∠A and ∠B are vertically opposite and ∠A = 2/3 ∠B, find ∠A.", "options": {"A": "72°", "B": "90°", "C": "108°", "D": "120°"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "Lines intersect at O. ∠AOB = (5x - 10)°, ∠COD = (3x + 30)°. Find ∠AOD.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "D"},
    {"id": 36, "difficulty": "medium", "question": "Two lines intersect forming angles. If one angle is 15° more than its vertically opposite angle, find the angles.", "options": {"A": "82.5° and 97.5°", "B": "75° and 90°", "C": "They must be equal", "D": "No solution"}, "correct_answer": "C"},
    {"id": 37, "difficulty": "medium", "question": "If the sum of two vertically opposite angles is 200°, what is each angle?", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "120°"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "In the figure, ∠AOC = 70°. Find ∠BOD + ∠COB.", "options": {"A": "180°", "B": "200°", "C": "220°", "D": "250°"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "Two lines intersect. The measures of three angles are 40°, x, and 140°. Find x.", "options": {"A": "40°", "B": "90°", "C": "100°", "D": "140°"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "If ∠P and ∠Q are vertically opposite and 2∠P = 3∠Q, find ∠P.", "options": {"A": "72°", "B": "90°", "C": "108°", "D": "120°"}, "correct_answer": "C"},
    {"id": 41, "difficulty": "medium", "question": "Lines intersect at point M. ∠PMQ = (4x + 20)°, ∠RMS = (6x - 40)°. Find x.", "options": {"A": "15", "B": "20", "C": "25", "D": "30"}, "correct_answer": "D"},
    {"id": 42, "difficulty": "medium", "question": "Two lines intersect. If one angle is twice its adjacent angle, find the vertically opposite angles.", "options": {"A": "60° and 120°", "B": "50° and 130°", "C": "45° and 135°", "D": "30° and 150°"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "medium", "question": "In the figure, if ∠1 = 65° and ∠2 = 115°, which angles are vertically opposite?", "options": {"A": "∠1 and ∠2", "B": "∠1 and ∠3", "C": "∠2 and ∠4", "D": "∠3 and ∠4"}, "correct_answer": "B"},
    {"id": 44, "difficulty": "medium", "question": "If two lines intersect and form an angle of (3x - 20)° and its vertically opposite angle is (x + 60)°, find x.", "options": {"A": "30", "B": "40", "C": "50", "D": "60"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "Lines AB and CD intersect at O. ∠AOC = 2∠BOD. Find ∠AOC.", "options": {"A": "60°", "B": "90°", "C": "120°", "D": "150°"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "Two lines intersect. The ratio of a pair of vertically opposite angles is 5:7. Find the angles.", "options": {"A": "75° and 105°", "B": "80° and 112°", "C": "90° and 126°", "D": "Vertically opposite angles must be equal"}, "correct_answer": "D"},
    {"id": 47, "difficulty": "medium", "question": "If ∠X and ∠Y are vertically opposite and ∠X = 3∠Y - 40°, find ∠X.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "B"},
    {"id": 48, "difficulty": "medium", "question": "In the figure, lines intersect at O. ∠AOB = 50°. Find the reflex angle AOB.", "options": {"A": "130°", "B": "230°", "C": "310°", "D": "360°"}, "correct_answer": "C"},
    {"id": 49, "difficulty": "medium", "question": "Two lines intersect. The sum of two vertically opposite angles is 240°. Find each angle.", "options": {"A": "100°", "B": "110°", "C": "120°", "D": "130°"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "medium", "question": "If ∠A = (2x + 30)° and its vertically opposite angle is (4x - 20)°, find the measure of the adjacent angle to ∠A.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "D"},
    {"id": 51, "difficulty": "medium", "question": "Lines PQ and RS intersect at T. ∠PTR = 70°. Find ∠QTS + ∠STR.", "options": {"A": "180°", "B": "200°", "C": "220°", "D": "250°"}, "correct_answer": "D"},
    {"id": 52, "difficulty": "medium", "question": "Two lines intersect forming angles in the ratio 2:3:2:3. Which pairs are vertically opposite?", "options": {"A": "2:2 and 3:3", "B": "2:3 and 2:3", "C": "All pairs", "D": "None"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "If the difference between a pair of vertically opposite angles is 0°, what is each angle?", "options": {"A": "0°", "B": "45°", "C": "90°", "D": "Any equal value"}, "correct_answer": "D"},
    {"id": 54, "difficulty": "medium", "question": "In the figure, ∠AOC = 40°. Find ∠BOD + ∠COB + ∠DOA.", "options": {"A": "320°", "B": "360°", "C": "400°", "D": "440°"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "medium", "question": "Two lines intersect. One angle is 30° less than three times its vertically opposite angle. Find the angles.", "options": {"A": "45° and 45°", "B": "52.5° and 52.5°", "C": "They must be equal", "D": "60° and 60°"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "If ∠P = (5x - 10)° and ∠Q = (3x + 50)° are vertically opposite, find the supplementary angle of ∠P.", "options": {"A": "70°", "B": "80°", "C": "90°", "D": "100°"}, "correct_answer": "D"},
    {"id": 57, "difficulty": "medium", "question": "Lines intersect at O. ∠AOB = 2x + 10, ∠COD = 3x - 20. Find the complement of x.", "options": {"A": "60°", "B": "70°", "C": "80°", "D": "90°"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "medium", "question": "Two lines intersect. The measures are y, 2y, y, and 2y. Which are vertically opposite pairs?", "options": {"A": "y and 2y", "B": "y and y, 2y and 2y", "C": "All pairs", "D": "None"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "medium", "question": "If ∠X and ∠Y are vertically opposite and ∠X + ∠Y = 180°, what is each angle?", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "180°"}, "correct_answer": "C"},
    {"id": 60, "difficulty": "medium", "question": "In the figure, lines intersect. ∠1 = 55°, ∠2 = 125°, ∠3 = 55°, ∠4 = 125°. Which statement is true?", "options": {"A": "∠1 and ∠2 are vertically opposite", "B": "∠1 and ∠3 are adjacent", "C": "∠1 and ∠3 are vertically opposite", "D": "∠2 and ∠4 are adjacent"}, "correct_answer": "C"},
    {"id": 61, "difficulty": "medium", "question": "Two lines intersect at right angles. What are the vertically opposite angles?", "options": {"A": "45° each", "B": "60° each", "C": "90° each", "D": "180° each"}, "correct_answer": "C"},
    {"id": 62, "difficulty": "medium", "question": "If ∠A = (x + 20)° and ∠B = (3x - 40)° are vertically opposite, find the value of 2x.", "options": {"A": "60", "B": "80", "C": "100", "D": "120"}, "correct_answer": "D"},
    {"id": 63, "difficulty": "medium", "question": "Lines intersect. One pair of vertically opposite angles are complementary. Find each angle.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "If the sum of all four angles formed by two intersecting lines is 360°, and one angle is 80°, find its vertically opposite angle.", "options": {"A": "80°", "B": "100°", "C": "120°", "D": "140°"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Two lines intersect. ∠1 = 70°. Find ∠3 if ∠1 and ∠3 are vertically opposite.", "options": {"A": "70°", "B": "110°", "C": "20°", "D": "160°"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "In the figure, if ∠AOB = 30° and lines are perpendicular, find ∠COD.", "options": {"A": "30°", "B": "60°", "C": "90°", "D": "150°"}, "correct_answer": "A"},
    {"id": 67, "difficulty": "medium", "question": "If ∠P and ∠Q are vertically opposite and 5∠P = 4∠Q, find ∠P.", "options": {"A": "80°", "B": "90°", "C": "100°", "D": "Vertically opposite angles must be equal"}, "correct_answer": "D"},
    {"id": 68, "difficulty": "medium", "question": "Lines intersect. One angle is (2x + 10)° and its vertically opposite is (x + 70)°. Find the adjacent angle to these.", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "D"},
    {"id": 69, "difficulty": "medium", "question": "Two lines intersect at O. ∠AOC = 50°. Find ∠BOD if they are vertically opposite.", "options": {"A": "40°", "B": "50°", "C": "130°", "D": "140°"}, "correct_answer": "B"},
    {"id": 70, "difficulty": "medium", "question": "If ∠X = 3y + 20 and ∠Y = 5y - 40 are vertically opposite, find y.", "options": {"A": "20", "B": "30", "C": "40", "D": "50"}, "correct_answer": "B"},
    {"id": 71, "difficulty": "medium", "question": "Lines intersect forming angles 40°, 140°, 40°, 140°. How many pairs of vertically opposite angles?", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 72, "difficulty": "medium", "question": "If two lines intersect and one angle is 1/3 of its vertically opposite angle, find the angles.", "options": {"A": "30° and 90°", "B": "45° and 135°", "C": "They must be equal", "D": "60° and 180°"}, "correct_answer": "C"},
    {"id": 73, "difficulty": "medium", "question": "In the figure, ∠AOD = 120°. Find ∠BOC if they are vertically opposite.", "options": {"A": "60°", "B": "120°", "C": "240°", "D": "300°"}, "correct_answer": "B"},
    {"id": 74, "difficulty": "medium", "question": "Two lines intersect. The measures are (x + 10)°, (2x - 20)°, (x + 10)°, (2x - 20)°. Find x.", "options": {"A": "30", "B": "40", "C": "50", "D": "60"}, "correct_answer": "C"},
    {"id": 75, "difficulty": "medium", "question": "If ∠A and ∠B are vertically opposite and ∠A + ∠B = 120°, what is each angle?", "options": {"A": "30°", "B": "40°", "C": "60°", "D": "120°"}, "correct_answer": "C"},
    {"id": 76, "difficulty": "medium", "question": "Lines intersect at point K. ∠LKM = 85°. Find ∠NKP if they are vertically opposite.", "options": {"A": "85°", "B": "95°", "C": "5°", "D": "175°"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Two lines intersect. One angle is 20° more than half its vertically opposite angle. Find the angles.", "options": {"A": "40° and 40°", "B": "50° and 50°", "C": "60° and 60°", "D": "They must be equal"}, "correct_answer": "D"},
    {"id": 78, "difficulty": "medium", "question": "If ∠P = 4x - 15 and ∠Q = 2x + 45 are vertically opposite, find the supplement of ∠P.", "options": {"A": "65°", "B": "75°", "C": "85°", "D": "95°"}, "correct_answer": "D"},
    {"id": 79, "difficulty": "medium", "question": "In the figure, lines intersect. ∠1 = 75°. Find ∠3 if they are vertically opposite.", "options": {"A": "15°", "B": "75°", "C": "105°", "D": "285°"}, "correct_answer": "B"},
    {"id": 80, "difficulty": "medium", "question": "Two lines intersect at right angles. What can you say about vertically opposite angles?", "options": {"A": "They are 45°", "B": "They are 90°", "C": "They are 180°", "D": "They vary"}, "correct_answer": "B"},

    {"id": 81, "difficulty": "hard", "question": "Three lines intersect at a point forming six angles. If one angle is 40°, find all angles assuming lines are distinct.", "options": {"A": "40°, 140°, 40°, 140°, 40°, 140°", "B": "40°, 40°, 100°, 100°, 40°, 40°", "C": "40°, 140°, 40°, 140°, 80°, 100°", "D": "Cannot be determined uniquely"}, "correct_answer": "D"},
    {"id": 82, "difficulty": "hard", "question": "In the figure, lines AB, CD, and EF intersect at O. If ∠AOC = 50° and ∠BOE = 70°, find ∠DOF.", "options": {"A": "50°", "B": "60°", "C": "70°", "D": "80°"}, "correct_answer": "B"},
    {"id": 83, "difficulty": "hard", "question": "Two lines intersect. If one angle is 30° more than twice its vertically opposite angle, are the lines perpendicular?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "B"},
    {"id": 84, "difficulty": "hard", "question": "Lines AB and CD intersect at O. Ray OE bisects ∠AOC. If ∠AOD = 70°, find ∠BOE.", "options": {"A": "125°", "B": "145°", "C": "155°", "D": "165°"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "Three lines intersect at a point forming six angles. The angles are in arithmetic progression with common difference 10°. Find the smallest angle.", "options": {"A": "40°", "B": "45°", "C": "50°", "D": "55°"}, "correct_answer": "D"},
    {"id": 86, "difficulty": "hard", "question": "In the figure, lines PQ and RS intersect at T. Ray TM bisects ∠PTR. If ∠QTS = 80°, find ∠MTQ.", "options": {"A": "40°", "B": "50°", "C": "60°", "D": "70°"}, "correct_answer": "D"},
    {"id": 87, "difficulty": "hard", "question": "Two lines intersect. The bisectors of two adjacent angles are perpendicular. Find all angles.", "options": {"A": "45°, 135°, 45°, 135°", "B": "60°, 120°, 60°, 120°", "C": "90°, 90°, 90°, 90°", "D": "30°, 150°, 30°, 150°"}, "correct_answer": "C"},
    {"id": 88, "difficulty": "hard", "question": "Lines intersect at O. Ray OA and OB are opposite rays. Ray OC is such that ∠AOC = 60°. Find ∠BOC if lines are extended.", "options": {"A": "60°", "B": "120°", "C": "240°", "D": "300°"}, "correct_answer": "B"},
    {"id": 89, "difficulty": "hard", "question": "In the figure, AB and CD intersect at O. OE bisects ∠BOD. If ∠AOC = 40°, find ∠COE.", "options": {"A": "70°", "B": "110°", "C": "140°", "D": "160°"}, "correct_answer": "D"},
    {"id": 90, "difficulty": "hard", "question": "Two lines intersect. The sum of two vertically opposite angles is equal to the sum of the other two vertically opposite angles. What can you conclude?", "options": {"A": "Lines are perpendicular", "B": "All angles are equal", "C": "Sum is 180°", "D": "This is always true"}, "correct_answer": "D"},
    {"id": 91, "difficulty": "hard", "question": "Three lines intersect pairwise forming a triangle. If one interior angle of triangle is 50°, find the vertically opposite angles at vertices.", "options": {"A": "50°, 130°, 50°, 130°", "B": "60°, 120°, 60°, 120°", "C": "Depends on configuration", "D": "Cannot determine"}, "correct_answer": "C"},
    {"id": 92, "difficulty": "hard", "question": "Lines intersect at O. Ray OP bisects ∠AOB where ∠AOB = 100°. Find the angle between OP and the bisector of ∠AOC where ∠AOC = 80°.", "options": {"A": "10°", "B": "20°", "C": "30°", "D": "40°"}, "correct_answer": "B"},
    {"id": 93, "difficulty": "hard", "question": "Two lines intersect. If the bisectors of a pair of vertically opposite angles are perpendicular, find the angles.", "options": {"A": "45°", "B": "60°", "C": "90°", "D": "120°"}, "correct_answer": "C"},
    {"id": 94, "difficulty": "hard", "question": "In the figure, AB ⊥ CD at O. Ray OE is such that ∠AOE = 30°. Find the reflex angle COE.", "options": {"A": "210°", "B": "240°", "C": "300°", "D": "330°"}, "correct_answer": "D"},
    {"id": 95, "difficulty": "hard", "question": "Three lines intersect at a point. The angles between them are 40°, 60°, and 80°. How many pairs of vertically opposite angles exist?", "options": {"A": "3", "B": "6", "C": "9", "D": "12"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Lines intersect. The ratio of a pair of adjacent angles is 3:5. Find the vertically opposite angles.", "options": {"A": "67.5° and 112.5°", "B": "60° and 120°", "C": "75° and 105°", "D": "45° and 135°"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "In the figure, lines AB, CD, EF intersect at O. ∠AOC = 2∠BOD and ∠COE = 3∠DOF. If ∠BOD = 30°, find ∠AOE.", "options": {"A": "120°", "B": "150°", "C": "180°", "D": "210°"}, "correct_answer": "D"},
    {"id": 98, "difficulty": "hard", "question": "Two lines intersect. The bisectors of two vertically opposite angles form an angle of 70°. Find the original angles.", "options": {"A": "35°", "B": "70°", "C": "110°", "D": "140°"}, "correct_answer": "D"},
    {"id": 99, "difficulty": "hard", "question": "Lines intersect at O. Ray OA bisects ∠BOC where ∠BOC = 120°. Find the angle between OA and the line bisecting ∠AOD where ∠AOD = 60°.", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "D"},
    {"id": 100, "difficulty": "hard", "question": "Three concurrent lines make angles of 20°, 50°, and 110° with each other. Find all vertically opposite angles formed.", "options": {"A": "20°, 160°, 50°, 130°, 110°, 70°", "B": "20°, 20°, 50°, 50°, 110°, 110°", "C": "20°, 70°, 50°, 130°, 110°, 70°", "D": "Cannot be uniquely determined"}, "correct_answer": "D"}
  ]
}