{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Angles Mastery & Calculation",
    "author": "Staples Education",
    "description": "Comprehensive check on angle pairs, algebraic angle systems, parallel-line transversals, polygon angle sums, and triangle angle relationships, from foundational to competitive difficulty.",
    "created": "2026-08-06T04:14:55.838Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "angles-mastery",
      "title": "Angles Mastery & Calculation",
      "blurb": "Angle pairs, parallel-line transversals, polygon angle sums, and triangle angle relationships — 30 questions from foundational to competitive.",
      "meta": "37 questions · easy → hard",
      "categories": [
        "angles",
        "parallel-perpendicular-lines",
        "triangles-review",
        "polygons"
      ],
      "questions": [
        {
          "id": "ang-001",
          "topic": "Angle Pairs",
          "difficulty": "easy",
          "stem": "Two angles are supplementary, and their measures are in the ratio 3:7. What is the measure of the smaller angle?",
          "choices": [
            "18°",
            "54°",
            "126°",
            "36°"
          ],
          "correctIndex": 1,
          "explanations": [
            "18° is the value of x, not an angle. Writing the angles as 3x and 7x gives 10x = 180, so x = 18 — but the question asks for an angle, which is 3x. Solving for the variable and stopping is the most common way to lose this question.",
            "Correct. Supplementary angles sum to 180°, so 3x + 7x = 180, giving x = 18. The smaller angle is 3(18) = 54°.",
            "126° is 7x — the larger angle. The ratio 3:7 puts the smaller measure on the 3 part, so this answers the opposite of what was asked.",
            "36° comes from treating the angles as complementary (summing to 90°): 10x = 90 would give x = 9 and 3x = 27, and 36° doesn't follow from either reading. Supplementary means a straight line, 180°, not a right angle."
          ],
          "hint": "Give the two angles a shared variable — 3x and 7x — and use what supplementary means about their sum.",
          "diagram": "<svg viewBox=\"0 0 240 110\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"20\" y1=\"90\" x2=\"220\" y2=\"90\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"90\" x2=\"60\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M 96 90 A 26 26 0 0 0 106 72\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><path d=\"M 150 90 A 32 32 0 0 0 140 66\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"84\" y=\"80\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3x</text><text x=\"152\" y=\"78\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">7x</text><circle cx=\"120\" cy=\"90\" r=\"3\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "The two angles form a straight line. Not drawn to scale"
        },
        {
          "id": "ang-002",
          "topic": "Angle Pairs",
          "difficulty": "medium",
          "stem": "The measure of an angle is 24° more than twice the measure of its complement. Find the measure of the angle.",
          "choices": [
            "22°",
            "44°",
            "68°",
            "112°"
          ],
          "correctIndex": 2,
          "explanations": [
            "22° is the complement of the correct answer (90 − 68). It answers a question that wasn't asked — the angle itself was requested, not its complement.",
            "44° results from dropping the 24°: solving x = 2(90 − x) gives 3x = 180 and x = 60, and 44° doesn't follow from that either. The '24° more than' has to be added to the doubled complement.",
            "Correct. The complement is 90 − x, so x = 2(90 − x) + 24. That expands to x = 204 − 2x, giving 3x = 204 and x = 68°.",
            "112° comes from using the supplement instead of the complement: x = 2(180 − x) + 24 heads toward a value above 90°. Complementary angles sum to 90°, not 180°."
          ],
          "hint": "Complementary angles sum to 90°, so if the angle is x, its complement is 90 − x. Translate 'more than twice' directly into that expression."
        },
        {
          "id": "ang-003",
          "topic": "Angle Pairs",
          "difficulty": "easy",
          "stem": "Angles A and B form a linear pair. If m∠A = 4x + 12 and m∠B = 6x − 22, what is the value of x?",
          "choices": [
            "10",
            "19",
            "20",
            "22"
          ],
          "correctIndex": 1,
          "explanations": [
            "10 comes from setting the two expressions equal to each other (4x + 12 = 6x − 22 gives x = 17, not 10) or from a slip in combining terms. A linear pair is not a congruent pair — the angles add to 180°, they don't match.",
            "Correct. A linear pair sums to 180°: (4x + 12) + (6x − 22) = 180 simplifies to 10x − 10 = 180, so 10x = 190 and x = 19.",
            "20 is what you get by dropping the −10 and solving 10x = 200. The constants +12 and −22 combine to −10, which has to be moved across before dividing.",
            "22 comes from summing to 200 rather than 180, or from adding the constants as +34 instead of −10. Check the sign on the −22 when combining."
          ],
          "hint": "A linear pair sits on a straight line. Add the two expressions and set the total to what a straight line measures.",
          "diagram": "<svg viewBox=\"0 0 240 110\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"20\" y1=\"90\" x2=\"220\" y2=\"90\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"90\" x2=\"165\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"60\" y=\"80\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4x + 12</text><text x=\"170\" y=\"78\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">6x − 22</text><circle cx=\"120\" cy=\"90\" r=\"3\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "∠A and ∠B form a linear pair. Not drawn to scale"
        },
        {
          "id": "ang-004",
          "topic": "Angle Pairs",
          "difficulty": "medium",
          "stem": "Two lines intersect. One angle measures 5x − 18 and the vertical angle opposite it measures 3x + 14. What is the measure of one of the remaining obtuse angles?",
          "choices": [
            "16°",
            "62°",
            "118°",
            "128°"
          ],
          "correctIndex": 2,
          "explanations": [
            "16° is the value of x, not an angle measure. It has to be substituted back into one of the expressions before it means anything geometric.",
            "62° is the vertical angle pair itself: 3(16) + 14 = 62°. But 62° is acute, and the question asks for one of the remaining obtuse angles — the ones adjacent to it.",
            "Correct. Vertical angles are congruent, so 5x − 18 = 3x + 14 gives x = 16 and an angle of 62°. The adjacent angle forms a linear pair with it: 180° − 62° = 118°.",
            "128° would be the answer if the 62° angle were subtracted from a straight line incorrectly, or if x were taken as 18. 180 − 62 = 118, not 128."
          ],
          "hint": "Vertical angles are equal — that gives you x. Then remember the obtuse angles are the ones next to that pair, not the pair itself.",
          "diagram": "<svg viewBox=\"0 0 240 125\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"25\" x2=\"215\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"105\" x2=\"215\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"92\" y=\"30\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5x − 18</text><text x=\"92\" y=\"108\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3x + 14</text><text x=\"38\" y=\"70\" fill=\"#6b6b76\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"196\" y=\"70\" fill=\"#6b6b76\" font-size=\"14\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><circle cx=\"120\" cy=\"65\" r=\"3\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "The two marked angles are vertical angles; the ? angles are the obtuse pair. Not drawn to scale"
        },
        {
          "id": "ang-005",
          "topic": "Angle Pairs",
          "difficulty": "medium",
          "stem": "Ray BD bisects ∠ABC. If m∠ABD = x² − 6 and m∠DBC = 5x + 8, find the positive value of x.",
          "choices": [
            "2",
            "7",
            "8",
            "14"
          ],
          "correctIndex": 1,
          "explanations": [
            "2 is the absolute value of the negative root. Factoring gives (x − 7)(x + 2) = 0, so the roots are 7 and −2. A negative root is discarded here because it would produce a negative angle measure, but its sign can't simply be flipped.",
            "Correct. A bisector splits the angle into two equal parts, so x² − 6 = 5x + 8. Rearranging gives x² − 5x − 14 = 0, which factors to (x − 7)(x + 2) = 0. The positive root is x = 7.",
            "8 comes from a factoring slip — looking for numbers that multiply to −14 and add to −5 gives −7 and +2, not −8 and something. Substituting 8 gives 58 and 48, which aren't equal.",
            "14 is the constant from the quadratic, not a root. It appears in x² − 5x − 14 = 0 as the product of the roots, not as a solution itself."
          ],
          "hint": "A bisector produces two equal angles. Set the expressions equal and you'll get a quadratic — only one root can be an angle measure.",
          "diagram": "<svg viewBox=\"0 0 240 120\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"40\" y1=\"100\" x2=\"200\" y2=\"100\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"100\" x2=\"165\" y2=\"22\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"100\" x2=\"190\" y2=\"62\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"96\" y=\"92\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x² − 6</text><text x=\"104\" y=\"60\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5x + 8</text><text x=\"196\" y=\"20\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"198\" y=\"60\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">D</text><text x=\"206\" y=\"104\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text><text x=\"28\" y=\"104\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><circle cx=\"40\" cy=\"100\" r=\"3\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "Ray BD bisects ∠ABC. Not drawn to scale"
        },
        {
          "id": "ang-006",
          "topic": "Angles Around a Point",
          "difficulty": "medium",
          "stem": "Three rays from a common vertex form three adjacent angles that together make a full revolution. Their measures are consecutive multiples of 5. What is the measure of the largest angle?",
          "choices": [
            "115°",
            "120°",
            "125°",
            "130°"
          ],
          "correctIndex": 2,
          "explanations": [
            "115° is the smallest of the three angles (5n where n = 23), not the largest. The three measures are 115°, 120°, and 125°.",
            "120° is the middle angle. It's also the answer you'd get by assuming the three angles are equal (360 ÷ 3), which ignores the 'consecutive multiples' condition entirely.",
            "Correct. Write the angles as 5n, 5n + 5, and 5n + 10. A full revolution is 360°, so 15n + 15 = 360, giving n = 23. The largest is 5(23) + 10 = 125°.",
            "130° would follow if the angles were spaced by 10 rather than 5, or if the revolution were taken as 375°. Consecutive multiples of 5 differ by exactly 5."
          ],
          "hint": "Angles filling a complete turn about a point sum to 360°. Write the three consecutive multiples with one variable.",
          "diagram": "<svg viewBox=\"0 0 240 140\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"70\" r=\"3\" fill=\"#7c3aed\"/><line x1=\"120\" y1=\"70\" x2=\"220\" y2=\"70\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"70\" x2=\"55\" y2=\"15\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"70\" x2=\"70\" y2=\"125\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"158\" y=\"52\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5n</text><text x=\"74\" y=\"62\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5n + 5</text><text x=\"128\" y=\"116\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">5n + 10</text></svg>",
          "diagramCaption": "Three angles about a single point. Not drawn to scale"
        },
        {
          "id": "ang-007",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "medium",
          "stem": "Lines l and m are parallel, cut by transversal t. A pair of same-side interior angles measure 7x + 16 and 4x − 1. What is the measure of the smaller angle?",
          "choices": [
            "15°",
            "59°",
            "121°",
            "60°"
          ],
          "correctIndex": 1,
          "explanations": [
            "15° is the value of x. It still has to be substituted into one of the expressions to become an angle.",
            "Correct. Same-side interior angles between parallel lines are supplementary: (7x + 16) + (4x − 1) = 180 gives 11x + 15 = 180, so x = 15. The angles are 121° and 59°, and the smaller is 59°.",
            "121° is the larger angle, 7(15) + 16. The question asks for the smaller of the pair.",
            "60° comes from treating the two angles as congruent instead of supplementary: 7x + 16 = 4x − 1 gives a negative x. Same-side interior angles are supplementary — it's the alternate interior pair that's congruent."
          ],
          "hint": "Same-side interior angles sit on the same side of the transversal, between the parallels. Decide whether that makes them equal or supplementary before writing an equation.",
          "diagram": "<svg viewBox=\"0 0 240 140\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"15\" y1=\"35\" x2=\"225\" y2=\"35\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"15\" y1=\"105\" x2=\"225\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"55\" y1=\"15\" x2=\"175\" y2=\"125\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"78\" y=\"55\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">7x + 16</text><text x=\"96\" y=\"98\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4x − 1</text><text x=\"226\" y=\"39\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">l</text><text x=\"226\" y=\"109\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">m</text><text x=\"178\" y=\"130\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">t</text></svg>",
          "diagramCaption": "l ∥ m, cut by transversal t. Not drawn to scale"
        },
        {
          "id": "ang-008",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "hard",
          "stem": "Two horizontal lines are parallel. A path bends between them, forming a zigzag. The upper segment makes a 35° angle with the top line and the lower segment makes a 45° angle with the bottom line. What is the measure of the bend angle z?",
          "choices": [
            "70°",
            "80°",
            "100°",
            "280°"
          ],
          "correctIndex": 1,
          "explanations": [
            "70° doubles the 35° instead of combining both given angles. Only one of the two deflections is 35°; the other is 45°.",
            "Correct. Draw an auxiliary line through the vertex parallel to both lines. It splits z into two parts, each an alternate interior angle with one of the given angles. So z = 35° + 45° = 80°.",
            "100° treats one of the angles as supplementary before adding — for example 180 − 35 − 45. The auxiliary line makes each part directly equal to a given angle, with no subtraction involved.",
            "280° is the reflex angle on the outside of the bend, 360° − 80°. The angle marked z is the interior one at the bend."
          ],
          "hint": "Draw a line through the bend parallel to the other two. Each half of z then pairs with one of the given angles.",
          "diagram": "<svg viewBox=\"0 0 240 135\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"15\" y1=\"25\" x2=\"225\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"15\" y1=\"115\" x2=\"225\" y2=\"115\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><polyline points=\"20,25 150,70 20,115\" fill=\"none\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linejoin=\"round\"/><text x=\"38\" y=\"42\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">35°</text><text x=\"38\" y=\"108\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">45°</text><text x=\"156\" y=\"74\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">z</text><circle cx=\"150\" cy=\"70\" r=\"3\" fill=\"#1f2024\"/></svg>",
          "diagramCaption": "The two horizontal lines are parallel. Not drawn to scale"
        },
        {
          "id": "ang-009",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "medium",
          "stem": "A pair of alternate exterior angles formed by a transversal across parallel lines measure 3x + 40 and 5x − 10. What is the measure of the acute angle adjacent to them?",
          "choices": [
            "25°",
            "65°",
            "115°",
            "50°"
          ],
          "correctIndex": 1,
          "explanations": [
            "25° is the value of x, not an angle measure.",
            "Correct. Alternate exterior angles are congruent: 3x + 40 = 5x − 10 gives x = 25, so each measures 115°. The adjacent angle forms a linear pair with it: 180° − 115° = 65°.",
            "115° is the alternate exterior angle itself. The question asks for the acute angle beside it, and 115° is obtuse.",
            "50° is 2x, which appears while solving (2x = 50) but is not an angle in the figure."
          ],
          "hint": "Alternate exterior angles are congruent — that gives x. Then use a linear pair to step to the adjacent angle."
        },
        {
          "id": "ang-010",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "The interior angles of a convex pentagon are x, 2x, 1.5x, 1.8x, and 2.7x. What is the measure of the smallest interior angle?",
          "choices": [
            "60°",
            "67.5°",
            "90°",
            "54°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A pentagon's interior angles sum to (5 − 2) × 180 = 540°. The coefficients total 9, so 9x = 540 and x = 60°. The smallest angle is x itself.",
            "67.5° comes from dividing 540 by 8 instead of 9. Adding the coefficients 1 + 2 + 1.5 + 1.8 + 2.7 gives 9.",
            "90° would follow from using 720° as the pentagon's sum (the hexagon value) — 720 ÷ 8 = 90. A pentagon has n = 5, so (n − 2) × 180 = 540°.",
            "54° divides 540 by 10 rather than 9. It's also close enough to the correct answer to slip past a quick check, so re-add the coefficients."
          ],
          "hint": "Use (n − 2) × 180 for the total, then add the coefficients of x before dividing."
        },
        {
          "id": "ang-011",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "Each interior angle of a regular convex polygon measures 168°. How many sides does the polygon have?",
          "choices": [
            "20",
            "24",
            "30",
            "36"
          ],
          "correctIndex": 2,
          "explanations": [
            "20 would correspond to an exterior angle of 18°, giving an interior angle of 162°, not 168°.",
            "24 corresponds to an exterior angle of 15° and an interior angle of 165°. Close, but the exterior angle here is 12°.",
            "Correct. Each exterior angle is 180° − 168° = 12°. Exterior angles of any convex polygon sum to 360°, so n = 360 ÷ 12 = 30.",
            "36 would come from dividing 360 by 10. The exterior angle is 12°, not 10°."
          ],
          "hint": "Go through the exterior angle. It's the supplement of the interior angle, and the exterior angles always total 360°."
        },
        {
          "id": "ang-012",
          "topic": "Polygon Angle Sums",
          "difficulty": "hard",
          "stem": "The ratio of an interior angle to an exterior angle of a regular polygon is 5:1. How many diagonals does the polygon have?",
          "choices": [
            "12",
            "54",
            "65",
            "90"
          ],
          "correctIndex": 1,
          "explanations": [
            "12 is the number of sides, not the number of diagonals. It's the halfway point of the problem — the diagonal formula still has to be applied.",
            "Correct. Interior and exterior angles are supplementary, so 5x + x = 180 gives x = 30° for the exterior angle. Then n = 360 ÷ 30 = 12 sides, and the number of diagonals is n(n − 3)/2 = 12(9)/2 = 54.",
            "65 comes from using n(n − 3)/2 with n = 13, or from n(n − 1)/2 with a slip. Each vertex connects to n − 3 others by a diagonal — not n − 1, since a vertex joins to itself and its two neighbours by sides, not diagonals.",
            "90 is n(n − 3) = 12 × 9 without halving. Every diagonal gets counted twice that way, once from each endpoint."
          ],
          "hint": "The ratio gives you the exterior angle, which gives you n. Then a separate formula turns n into a diagonal count."
        },
        {
          "id": "ang-013",
          "topic": "Triangle Angle Relationships",
          "difficulty": "medium",
          "stem": "In △ABC, the exterior angle at vertex C measures 142°. If m∠A = 3x + 10 and m∠B = 5x − 4, what is m∠A?",
          "choices": [
            "17",
            "61°",
            "81°",
            "38°"
          ],
          "correctIndex": 1,
          "explanations": [
            "17 is the value of x. It carries no degree sign because it isn't an angle — substituting it into 3x + 10 is the step that's missing.",
            "Correct. The Exterior Angle Theorem says the exterior angle equals the sum of the two remote interior angles: (3x + 10) + (5x − 4) = 142 gives 8x + 6 = 142, so x = 17 and m∠A = 3(17) + 10 = 61°.",
            "81° is m∠B, which is 5(17) − 4. The question asks for m∠A.",
            "38° is the interior angle at C, the supplement of 142°. That's a real angle in the figure, but it isn't ∠A."
          ],
          "hint": "The exterior angle equals the sum of the two interior angles not adjacent to it — no need to find the third angle first.",
          "diagram": "<svg viewBox=\"0 0 240 130\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"30\" y1=\"105\" x2=\"175\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"175\" y1=\"105\" x2=\"225\" y2=\"105\" stroke=\"#6b6b76\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"30\" y1=\"105\" x2=\"110\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"110\" y1=\"25\" x2=\"175\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"40\" y=\"96\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3x + 10</text><text x=\"96\" y=\"44\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">B</text><text x=\"128\" y=\"96\" fill=\"#6b6b76\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"180\" y=\"96\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">142°</text><text x=\"20\" y=\"110\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">A</text><text x=\"172\" y=\"120\" fill=\"#6b6b76\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">C</text></svg>",
          "diagramCaption": "The exterior angle at C is 142°; m∠B = 5x − 4. Not drawn to scale"
        },
        {
          "id": "ang-014",
          "topic": "Triangle Angle Relationships",
          "difficulty": "easy",
          "stem": "An isosceles triangle has an apex angle of 76°. What is the measure of one of its base angles?",
          "choices": [
            "52°",
            "76°",
            "104°",
            "38°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The interior angles sum to 180°, and the two base angles of an isosceles triangle are equal. So 76 + 2b = 180 gives 2b = 104 and b = 52°.",
            "76° repeats the apex angle. In an isosceles triangle it's the two base angles that are equal to each other, not equal to the apex.",
            "104° is the combined measure of both base angles. It still has to be halved, since the question asks for one of them.",
            "38° halves the apex angle instead of the remaining 104°. The apex isn't split by anything here."
          ],
          "hint": "Subtract the apex from 180° first — what's left belongs to two equal angles.",
          "diagram": "<svg viewBox=\"0 0 240 125\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"45\" y1=\"105\" x2=\"195\" y2=\"105\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"45\" y1=\"105\" x2=\"120\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"195\" y1=\"105\" x2=\"120\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"108\" y=\"48\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">76°</text><text x=\"56\" y=\"98\" fill=\"#6b6b76\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text><text x=\"178\" y=\"98\" fill=\"#6b6b76\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
          "diagramCaption": "The two base angles are congruent. Not drawn to scale"
        },
        {
          "id": "ang-015",
          "topic": "Triangle Angle Relationships",
          "difficulty": "medium",
          "stem": "In a right triangle, the two acute angles differ by 28°. What is the measure of the larger acute angle?",
          "choices": [
            "31°",
            "59°",
            "62°",
            "74°"
          ],
          "correctIndex": 1,
          "explanations": [
            "31° is the smaller acute angle, 90 − 59. The question asks for the larger of the two.",
            "Correct. The acute angles of a right triangle sum to 90°. With x + y = 90 and x − y = 28, adding the equations gives 2x = 118, so x = 59°.",
            "62° comes from using 180° as the sum of the two acute angles instead of 90°: (180 + 28)/2 = 104 is too large, and 62° follows from a related slip. The right angle already uses 90° of the triangle's 180°.",
            "74° is (120 + 28)/2, or the result of splitting the difference against the wrong total. Check that the two acute angles add to 90°."
          ],
          "hint": "The right angle takes 90° of the triangle's 180°, so the two acute angles share what's left."
        },
        {
          "id": "ang-016",
          "topic": "Triangle Angle Relationships",
          "difficulty": "hard",
          "stem": "A dart (arrowhead) quadrilateral has three outer angles measuring 32°, 28°, and 35°. What is the measure of the reflex-forming interior angle x at the indented vertex?",
          "choices": [
            "63°",
            "95°",
            "85°",
            "72°"
          ],
          "correctIndex": 1,
          "explanations": [
            "63° adds only two of the three outer angles (28 + 35). All three contribute to the angle at the indented vertex.",
            "Correct. In a dart, the angle at the indented vertex equals the sum of the three remote outer angles: 32° + 28° + 35° = 95°. This follows from splitting the dart into two triangles with a line from the apex through the indented vertex.",
            "85° is the result of an arithmetic slip — 32 + 28 + 35 is 95, not 85. Re-add the three given angles.",
            "72° subtracts rather than adds somewhere, for example 180 − (32 + 28 + 35 + 13). The three outer angles combine additively here."
          ],
          "hint": "Draw a segment from the top vertex through the indented one. It splits the figure into two triangles, and the exterior angle relationship does the rest.",
          "diagram": "<svg viewBox=\"0 0 240 135\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><path d=\"M120,20 L50,120 L120,88 L190,120 Z\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><text x=\"126\" y=\"44\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">32°</text><text x=\"66\" y=\"112\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">28°</text><text x=\"150\" y=\"112\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">35°</text><text x=\"113\" y=\"80\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text></svg>",
          "diagramCaption": "A dart: the vertex marked x points inward. Not drawn to scale"
        },
        {
          "id": "ang-017",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "hard",
          "stem": "Two parallel lines are cut by a transversal. One angle measures 4x² + 20 and its corresponding angle measures 20x + 4. Both solutions for x are positive — what is the greater one?",
          "choices": [
            "2",
            "4",
            "5",
            "8"
          ],
          "correctIndex": 1,
          "explanations": [
            "2 isn't a root. Substituting it gives 36 and 44, which aren't equal, so the corresponding-angle condition fails.",
            "Correct. Corresponding angles are congruent: 4x² + 20 = 20x + 4 rearranges to 4x² − 20x + 16 = 0, or x² − 5x + 4 = 0, which factors to (x − 4)(x − 1) = 0. The roots are 4 and 1, and the greater is 4.",
            "5 is the sum of the two roots (4 + 1), which appears as the middle coefficient of x² − 5x + 4. The sum of the roots isn't itself a root.",
            "8 comes from solving 4x² − 20x + 16 = 0 without dividing through by 4 first, then mis-factoring. Dividing by 4 gives the clean x² − 5x + 4."
          ],
          "hint": "Corresponding angles are congruent. Set the expressions equal, collect everything on one side, and divide out the common factor before factoring."
        },
        {
          "id": "ang-018",
          "topic": "Polygon Angle Sums",
          "difficulty": "easy",
          "stem": "What is the sum of the exterior angles of a convex nonagon, taking one exterior angle at each vertex?",
          "choices": [
            "360°",
            "540°",
            "1260°",
            "180°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The exterior angles of any convex polygon sum to 360°, no matter how many sides it has. Walking once around the polygon turns you through one complete revolution.",
            "540° is the interior angle sum of a pentagon, not an exterior sum for anything.",
            "1260° is the interior angle sum of a nonagon: (9 − 2) × 180. This is the interior total, and the question asks about exterior angles.",
            "180° is the interior angle sum of a triangle. Exterior angles complete a full turn, which is twice this."
          ],
          "hint": "Picture walking the perimeter once and returning to your starting heading. How far have you turned in total?"
        },
        {
          "id": "ang-019",
          "topic": "Angles Around a Point",
          "difficulty": "medium",
          "stem": "Three adjacent angles lie along a straight line. The second is double the first, and the third is triple the second. What is the measure of the second angle?",
          "choices": [
            "20°",
            "40°",
            "60°",
            "120°"
          ],
          "correctIndex": 1,
          "explanations": [
            "20° is the first angle, x. The question asks for the second, which is 2x.",
            "Correct. Let the first be x. The second is 2x and the third is 3(2x) = 6x. Along a straight line they sum to 180°, so 9x = 180 and x = 20. The second angle is 2(20) = 40°.",
            "60° would follow from reading the third angle as 3x rather than triple the second. 'Triple the second' means 3 × 2x = 6x, giving coefficients 1, 2, 6 rather than 1, 2, 3.",
            "120° is the third angle, 6x. It's the largest of the three, not the middle one."
          ],
          "hint": "Build the third angle from the second, not from the first — then use the straight-line total.",
          "diagram": "<svg viewBox=\"0 0 240 120\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"20\" y1=\"100\" x2=\"220\" y2=\"100\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"100\" x2=\"55\" y2=\"40\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"100\" x2=\"120\" y2=\"25\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"100\" x2=\"190\" y2=\"45\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"76\" y=\"92\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text><text x=\"110\" y=\"78\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">2x</text><text x=\"150\" y=\"88\" fill=\"#1f2024\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">6x</text><circle cx=\"120\" cy=\"100\" r=\"3\" fill=\"#7c3aed\"/></svg>",
          "diagramCaption": "Three adjacent angles on a straight line. Not drawn to scale"
        },
        {
          "id": "ang-020",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "medium",
          "stem": "A horizontal cross-strut runs parallel to the base of a triangular frame. The base angle at the left is 58°. What is the measure of the corresponding angle x where the left side meets the strut?",
          "choices": [
            "58°",
            "122°",
            "62°",
            "42°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The strut is parallel to the base, and the left side of the frame is a transversal crossing both. Angle x and the 58° base angle are corresponding angles, so they're congruent: x = 58°.",
            "122° is the supplement of 58°. That would be the angle on the other side of the strut — the one forming a linear pair with x, not x itself.",
            "62° comes from treating the angles as complementary (90 − 58 = 32) or from an unrelated subtraction. Corresponding angles across parallel lines are equal, with no subtraction involved.",
            "42° doesn't follow from any relationship in the figure. Check which pair of angles the transversal actually creates."
          ],
          "hint": "The strut and the base are parallel, and one side of the frame crosses both. Identify which angle pair that makes.",
          "diagram": "<svg viewBox=\"0 0 240 130\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"110\" x2=\"215\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"110\" x2=\"120\" y2=\"25\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"215\" y1=\"110\" x2=\"120\" y2=\"25\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"68\" y1=\"72\" x2=\"172\" y2=\"72\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"40\" y=\"103\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">58°</text><text x=\"76\" y=\"66\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text></svg>",
          "diagramCaption": "The strut is parallel to the base. Not drawn to scale"
        },
        {
          "id": "ang-021",
          "topic": "Triangle Angle Relationships",
          "difficulty": "medium",
          "stem": "A triangle has angles 2x − 5, 3x + 10, and 4x − 5. How is the triangle classified by its angles?",
          "choices": [
            "Acute and right",
            "Right triangle",
            "Obtuse triangle",
            "Acute triangle"
          ],
          "correctIndex": 3,
          "explanations": [
            "No triangle can be both acute and right — the terms describe the same thing, the largest angle, and they contradict each other.",
            "Right would require one angle of exactly 90°. The angles here are 35°, 70°, and 75°; none is 90°.",
            "Obtuse would require an angle greater than 90°. The largest here is 75°.",
            "Correct. The angles sum to 180°: 9x = 180, so x = 20. That gives 35°, 70°, and 75°. Every angle is under 90°, so the triangle is acute."
          ],
          "hint": "Find the three actual measures first, then look only at the largest one — it decides the classification."
        },
        {
          "id": "ang-022",
          "topic": "Polygon Angle Sums",
          "difficulty": "easy",
          "stem": "What is the sum of the interior angles of a dodecagon (12 sides)?",
          "choices": [
            "1440°",
            "1800°",
            "2160°",
            "360°"
          ],
          "correctIndex": 1,
          "explanations": [
            "1440° uses n = 10: (10 − 2) × 180. A dodecagon has 12 sides, not 10.",
            "Correct. The interior angle sum is (n − 2) × 180. For n = 12 that's 10 × 180 = 1800°.",
            "2160° is 12 × 180, skipping the −2. The formula subtracts 2 because a polygon splits into n − 2 triangles, not n.",
            "360° is the sum of the exterior angles, which is 360° for every convex polygon. This question asks about interior angles."
          ],
          "hint": "A polygon with n sides divides into how many triangles? Multiply that count by 180°."
        },
        {
          "id": "ang-023",
          "topic": "Polygon Angle Sums",
          "difficulty": "easy",
          "stem": "Each exterior angle of a regular polygon measures 15°. How many vertices does it have?",
          "choices": [
            "12",
            "20",
            "24",
            "48"
          ],
          "correctIndex": 2,
          "explanations": [
            "12 would correspond to exterior angles of 30° (360 ÷ 12). Here each is 15°, which is half that, so the polygon has twice as many sides.",
            "20 would correspond to exterior angles of 18°.",
            "Correct. Exterior angles of a convex polygon sum to 360°, and in a regular polygon they're all equal. So n = 360 ÷ 15 = 24. A polygon has as many vertices as sides.",
            "48 comes from dividing 720 by 15. The exterior angles total 360°, not 720°."
          ],
          "hint": "The exterior angles are all equal and together make one full turn."
        },
        {
          "id": "ang-024",
          "topic": "Angle Pairs",
          "difficulty": "easy",
          "stem": "A vertical ray is perpendicular to a horizontal line at a point. Another ray from that same point makes a 38° angle with the horizontal line. What is the measure of the angle x between that ray and the vertical ray?",
          "choices": [
            "38°",
            "52°",
            "62°",
            "48°"
          ],
          "correctIndex": 1,
          "explanations": [
            "38° repeats the given angle. The two angles here are different parts of the right angle, not equal to each other.",
            "Correct. The vertical and horizontal rays form a 90° angle, and the two smaller angles split it. They're complementary: x = 90° − 38° = 52°.",
            "62° subtracts from 100 rather than 90. A perpendicular forms exactly 90°.",
            "48° subtracts from 86, or reflects an arithmetic slip. 90 − 38 = 52."
          ],
          "hint": "Perpendicular means exactly 90°. The two angles shown divide that right angle between them.",
          "diagram": "<svg viewBox=\"0 0 240 130\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"110\" x2=\"215\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"110\" x2=\"120\" y2=\"22\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><path d=\"M120,96 L134,96 L134,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><line x1=\"120\" y1=\"110\" x2=\"45\" y2=\"45\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"62\" y=\"100\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">38°</text><text x=\"96\" y=\"62\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text></svg>",
          "diagramCaption": "The vertical ray is perpendicular to the horizontal line. Not drawn to scale"
        },
        {
          "id": "ang-025",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "The interior angle of a regular polygon is given by (8 × 180) ÷ 10. What is the measure of one exterior angle?",
          "choices": [
            "144°",
            "36°",
            "72°",
            "18°"
          ],
          "correctIndex": 1,
          "explanations": [
            "144° is the interior angle itself — the value the given expression evaluates to. The exterior angle is its supplement.",
            "Correct. The expression is (n − 2) × 180 ÷ n with n = 10, giving an interior angle of 144°. The exterior angle is the supplement: 180° − 144° = 36°.",
            "72° is double the exterior angle, or 360 ÷ 5. The polygon here has 10 sides, not 5.",
            "18° is 360 ÷ 20, which would describe a 20-sided polygon. The denominator 10 in the given expression is n."
          ],
          "hint": "Read n straight off the expression, then remember interior and exterior angles at a vertex form a straight line."
        },
        {
          "id": "ang-026",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "medium",
          "stem": "Two parallel lines are cut by a transversal. A pair of alternate interior angles measure 8x − 40 and 4x + 16. What is the measure of each of these angles?",
          "choices": [
            "14°",
            "72°",
            "108°",
            "56°"
          ],
          "correctIndex": 1,
          "explanations": [
            "14° is the value of x, which still has to be substituted into one of the expressions.",
            "Correct. Alternate interior angles are congruent: 8x − 40 = 4x + 16 gives 4x = 56, so x = 14. Each angle measures 4(14) + 16 = 72°.",
            "108° is the supplement of 72°. That's the co-interior angle beside this pair, not the pair itself — alternate interior angles are equal, not supplementary.",
            "56° is 4x, a value that appears while solving but isn't an angle in the figure."
          ],
          "hint": "Alternate interior angles sit on opposite sides of the transversal. Decide whether that makes them equal or supplementary."
        },
        {
          "id": "ang-027",
          "topic": "Triangle Angle Relationships",
          "difficulty": "easy",
          "stem": "In a right triangle, one acute angle is four times the other. What is the measure of the smaller acute angle?",
          "choices": [
            "18°",
            "72°",
            "15°",
            "30°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The two acute angles sum to 90°, so x + 4x = 90 gives 5x = 90 and x = 18°.",
            "72° is the larger acute angle, 4x. The question asks for the smaller.",
            "15° comes from using 75° or 180° as the total in a way that mis-splits the ratio. The two acute angles of a right triangle share exactly 90°.",
            "30° would follow from x + 4x = 150, or from splitting 90° into thirds. The ratio 1:4 divides 90° into five parts, not three."
          ],
          "hint": "The right angle already accounts for 90°, so the two acute angles split the remaining 90° in a 1:4 ratio."
        },
        {
          "id": "ang-028",
          "topic": "Parallel Lines & Transversals",
          "difficulty": "easy",
          "stem": "Two parallel lines are cut by a transversal. One interior angle measures 112°. What is the measure of the co-interior (same-side interior) angle x?",
          "choices": [
            "112°",
            "68°",
            "78°",
            "88°"
          ],
          "correctIndex": 1,
          "explanations": [
            "112° would be correct for the alternate interior or corresponding angle, which are congruent to the given one. Same-side interior angles are supplementary instead.",
            "Correct. Co-interior angles between parallel lines are supplementary: x = 180° − 112° = 68°.",
            "78° subtracts from 190 rather than 180. A straight line is 180°.",
            "88° subtracts from 200, or reflects an arithmetic slip. 180 − 112 = 68."
          ],
          "hint": "Same-side interior angles are on the same side of the transversal. That side relationship makes them supplementary, not equal.",
          "diagram": "<svg viewBox=\"0 0 240 132\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"15\" y1=\"30\" x2=\"225\" y2=\"30\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"15\" y1=\"110\" x2=\"225\" y2=\"110\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"50\" y1=\"20\" x2=\"165\" y2=\"120\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"72\" y=\"50\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">112°</text><text x=\"112\" y=\"100\" fill=\"#1f2024\" font-size=\"13\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text></svg>",
          "diagramCaption": "The two horizontal lines are parallel. Not drawn to scale"
        },
        {
          "id": "ang-029",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "The interior angles of a convex polygon sum to 900°. How many sides does it have?",
          "choices": [
            "5",
            "6",
            "7",
            "8"
          ],
          "correctIndex": 2,
          "explanations": [
            "5 gives (5 − 2) × 180 = 540°, the pentagon's interior sum.",
            "6 gives (6 − 2) × 180 = 720°, the hexagon's interior sum.",
            "Correct. Set (n − 2) × 180 = 900. Dividing gives n − 2 = 5, so n = 7 — a heptagon.",
            "8 gives (8 − 2) × 180 = 1080°, which overshoots 900°."
          ],
          "hint": "Divide the total by 180° first — what's left is the number of triangles the polygon splits into, which is two fewer than the number of sides."
        },
        {
          "id": "ang-030",
          "topic": "Angles Around a Point",
          "difficulty": "medium",
          "stem": "Four rays from a common point divide the plane into four angles measuring x, 2x + 10, 3x − 20, and 4x + 10. What is the measure of the largest angle?",
          "choices": [
            "36°",
            "82°",
            "124°",
            "154°"
          ],
          "correctIndex": 3,
          "explanations": [
            "36° is the value of x, which is also the smallest angle here — but it's the value of the variable, and the question asks for the largest angle.",
            "82° is 2x + 10, the second angle. Compare all four measures before choosing.",
            "124° is 3x − 20, the third angle. The largest coefficient belongs to 4x + 10.",
            "Correct. A full turn is 360°. Summing gives x + (2x + 10) + (3x − 20) + (4x + 10) = 10x = 360, so x = 36. The largest angle is 4(36) + 10 = 154°."
          ],
          "hint": "The constants cancel when you add all four expressions. Angles filling a complete turn sum to 360°.",
          "diagram": "<svg viewBox=\"0 0 240 150\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><circle cx=\"120\" cy=\"72\" r=\"3\" fill=\"#7c3aed\"/><line x1=\"120\" y1=\"72\" x2=\"225\" y2=\"72\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"72\" x2=\"165\" y2=\"15\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"72\" x2=\"35\" y2=\"40\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"72\" x2=\"70\" y2=\"135\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><text x=\"160\" y=\"58\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">x</text><text x=\"104\" y=\"30\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">2x + 10</text><text x=\"30\" y=\"72\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">3x − 20</text><text x=\"122\" y=\"126\" fill=\"#1f2024\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">4x + 10</text></svg>",
          "diagramCaption": "Four angles filling a complete turn. Not drawn to scale"
        },
        {
          "id": "ang-031",
          "topic": "Complementary Angles",
          "difficulty": "medium",
          "stem": "Two angles are complementary and one is 27° larger than the other. Find the smaller angle.",
          "choices": [
            "31.5°",
            "58.5°",
            "76.5°",
            "63°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Complementary means the two sum to 90°. Let the smaller be x; then x + (x + 27) = 90, so 2x = 63 and x = 31.5°. The larger is 58.5°, and 31.5 + 58.5 = 90 as required.",
            "58.5° is the LARGER of the two angles, not the smaller. It is the correct pair, but the question asks for the other one.",
            "76.5° comes from treating the angles as supplementary: (180 − 27)/2 = 76.5. Complementary is 90°, not 180° — this is the single most common slip on this problem type.",
            "63° is 90 − 27, which subtracts the difference from the total in one step. That skips the fact that the difference has to be split between the two angles, not removed entirely."
          ],
          "hint": "Complementary fixes the sum. Name the smaller angle x and write the larger in terms of it."
        },
        {
          "id": "ang-032",
          "topic": "Supplement and Complement",
          "difficulty": "hard",
          "stem": "An angle's supplement is 4 times its complement. Find the angle.",
          "choices": [
            "30°",
            "45°",
            "60°",
            "72°"
          ],
          "correctIndex": 2,
          "explanations": [
            "30° gives a supplement of 150° and a complement of 60°. But 4 × 60 = 240, not 150, so the condition fails.",
            "45° gives a supplement of 135° and a complement of 45°. Then 4 × 45 = 180, not 135. This choice is tempting because 45° is self-complementary, but that is not what the problem asks.",
            "Correct. Let the angle be a. Its supplement is 180 − a and its complement is 90 − a, so 180 − a = 4(90 − a) = 360 − 4a. Then 3a = 180 and a = 60°. Check: supplement 120°, complement 30°, and 120 = 4 × 30.",
            "72° gives a supplement of 108° and a complement of 18°. Then 4 × 18 = 72, which is the angle itself rather than the supplement — a sign the two expressions got swapped."
          ],
          "hint": "Write both the supplement and the complement as expressions in the same unknown before setting up the equation."
        },
        {
          "id": "ang-033",
          "topic": "Co-interior Angles",
          "difficulty": "medium",
          "stem": "Lines ℓ ∥ m are cut by a transversal. A co-interior (same-side interior) pair measures (2x + 10)° and (3x − 30)°. Find x.",
          "choices": [
            "20",
            "40",
            "8",
            "44"
          ],
          "correctIndex": 1,
          "explanations": [
            "x = 20 comes from setting the two expressions EQUAL: 2x + 10 = 3x − 30 gives x = 40, not 20 — but students who mis-solve that equation often land here. Co-interior angles are not equal in the first place.",
            "Correct. Co-interior angles on the same side of a transversal between parallel lines are SUPPLEMENTARY. So (2x + 10) + (3x − 30) = 180, giving 5x − 20 = 180, then 5x = 200 and x = 40. Both angles work out to 90°.",
            "x = 8 comes from setting the sum to 90° instead of 180°. Co-interior angles are supplementary, not complementary — the pair straddles a straight line, not a right angle.",
            "x = 44 comes from solving 5x = 220, which drops the sign on the −20 when moving it across. The constant is −20, so it is added to 180, not subtracted."
          ],
          "hint": "Of the four transversal relationships, three give equality and one gives a sum of 180°. Identify which one same-side interior angles fall under."
        },
        {
          "id": "ang-034",
          "topic": "Vertical Angles",
          "difficulty": "easy",
          "stem": "Two vertical angles measure (5y − 12)° and (3y + 20)°. Find the measure of each angle.",
          "choices": [
            "16°",
            "68°",
            "112°",
            "84°"
          ],
          "correctIndex": 1,
          "explanations": [
            "16° is the value of y, not the angle. Solving 5y − 12 = 3y + 20 gives 2y = 32 and y = 16, but the question asks for the angle measure, which still needs substituting back.",
            "Correct. Vertical angles are equal, so 5y − 12 = 3y + 20, giving 2y = 32 and y = 16. Substituting back: 5(16) − 12 = 68°, and the check 3(16) + 20 = 68° agrees.",
            "112° is the SUPPLEMENT of the correct answer. It comes from treating the two angles as a linear pair summing to 180° rather than as vertical angles, which are equal.",
            "84° comes from substituting y = 16 into an incorrectly simplified expression. Both given expressions must evaluate to the same number — check one against the other before committing."
          ],
          "hint": "Vertical angles are equal. Solve for the variable first, then remember the question asks for the angle, not the variable."
        },
        {
          "id": "ang-035",
          "topic": "Angles on a Line",
          "difficulty": "medium",
          "stem": "Three angles lie along a straight line and measure x°, (2x)°, and (x + 40)°. Find the largest of the three.",
          "choices": [
            "35°",
            "70°",
            "75°",
            "105°"
          ],
          "correctIndex": 2,
          "explanations": [
            "35° is the value of x and the SMALLEST of the three angles. Solving x + 2x + (x + 40) = 180 gives 4x = 140 and x = 35, but that is only the first angle.",
            "70° is the middle angle, 2x = 2(35). Comparing all three — 35°, 70° and 75° — shows this is not the largest.",
            "Correct. The three angles form a straight angle, so x + 2x + (x + 40) = 180, giving 4x + 40 = 180, then 4x = 140 and x = 35. The three angles are 35°, 70° and 75°, so the largest is 75°.",
            "105° would come from using 360° as the total instead of 180°. Angles along a straight LINE sum to 180°; it is angles around a POINT that sum to 360°."
          ],
          "hint": "Angles that together form a straight line have a fixed sum. Find that sum first, then compare all three results."
        },
        {
          "id": "ang-036",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "Two parallel lines are cut by a transversal. A student sees a pair of co-interior (same-side interior) angles and marks them congruent. What is the correct relationship, and why?",
          "choices": [
            "They are congruent, because all angles formed by parallel lines are congruent.",
            "They are supplementary, summing to 180°, because they form a linear pair with a pair of congruent alternate interior angles.",
            "They are complementary, summing to 90°.",
            "There is no fixed relationship without knowing the angle of the transversal."
          ],
          "correctIndex": 1,
          "explanations": [
            "Only some of the pairs are congruent — corresponding, alternate interior and alternate exterior. Co-interior angles are the pair that is supplementary instead.",
            "Correct. Same-side interior angles sum to 180°. Congruence holds for corresponding and alternate pairs, but not for this one.",
            "Complementary means 90°, which does not arise from parallel lines and a transversal.",
            "The relationship is fixed by the lines being parallel, whatever angle the transversal makes."
          ],
          "hint": "Sort the transversal pairs into the ones that are congruent and the one that is supplementary.",
          "kind": "reasoning"
        },
        {
          "id": "ang-037",
          "topic": "Reasoning: What's Missing",
          "difficulty": "hard",
          "stem": "A diagram shows two lines cut by a transversal, with one pair of corresponding angles measuring 63° and 63°. What can be concluded, and by which reasoning?",
          "choices": [
            "Nothing — congruent corresponding angles can occur without the lines being parallel.",
            "The lines are parallel, by the converse of the Corresponding Angles Postulate.",
            "The lines are perpendicular, since 63° is not 90°.",
            "The lines must be the same line."
          ],
          "correctIndex": 1,
          "explanations": [
            "This gets the logic backwards. The CONVERSE is a theorem in its own right: congruent corresponding angles force the lines to be parallel.",
            "Correct. If two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel. Using a theorem in this direction is what a converse is for.",
            "Perpendicular lines meet at 90°. Nothing here indicates that, and 63° would rule it out.",
            "Two distinct parallel lines are not the same line — they simply never meet."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}