{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Polygon Angles, Area & Quadrilaterals",
    "author": "Staples Education",
    "description": "Interior and exterior angle sums for convex and regular polygons, the quadrilateral property chain, and area formulas for the rhombus, parallelogram, trapezoid and regular polygon — each explained by the dissection it comes from. Converted from a worked derivation board on rhombus and parallelogram area, and the wider tutoring whiteboard archive.",
    "created": "2026-08-06T04:14:56.204Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "polygon-area-quads",
      "title": "Polygon Angles, Area & Quadrilaterals",
      "blurb": "Interior and exterior angle sums, the quadrilateral property chain, and the area formulas for rhombus, parallelogram, trapezoid and regular polygon — each tied to the dissection it comes from.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "polygons",
        "quadrilaterals"
      ],
      "questions": [
        {
          "id": "pqa-001",
          "topic": "Regular Polygons",
          "difficulty": "medium",
          "stem": "What is the measure of ONE interior angle of a regular octagon (n = 8)?",
          "choices": [
            "45°",
            "135°",
            "140°",
            "1080°"
          ],
          "correctIndex": 1,
          "explanations": [
            "45° is one EXTERIOR angle, 360°/8. The interior angle is its supplement.",
            "Correct. Two routes give the same answer. Via the exterior angle: each is 360°/8 = 45°, so each interior angle is 180° − 45° = 135°. Or directly: (n − 2)·180°/n = (6)(180°)/8 = 1080°/8 = 135°.",
            "140° is the interior angle of a regular NONAGON (n = 9). For an octagon the value is 135°.",
            "1080° is the SUM of all eight interior angles, (8 − 2)(180°). The question asks for one of them, so divide by 8."
          ],
          "hint": "Go through the exterior angle — they always total 360°, whatever the polygon. Then take the supplement.",
          "diagram": "<svg viewBox=\"0 0 240 168\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"95,18 145,18 180,53 180,103 145,138 95,138 60,103 60,53\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><text x=\"146\" y=\"44\" fill=\"#1f2024\" 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=\"78\" y=\"158\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">regular octagon</text></svg>",
          "diagramCaption": "All eight interior angles are equal. Not drawn to scale"
        },
        {
          "id": "pqa-002",
          "topic": "Quadrilateral Properties",
          "difficulty": "medium",
          "stem": "Which quadrilateral has diagonals that are perpendicular and bisect each other, but are NOT necessarily congruent?",
          "choices": [
            "Parallelogram",
            "Rectangle",
            "Rhombus",
            "Isosceles trapezoid"
          ],
          "correctIndex": 2,
          "explanations": [
            "A general parallelogram has diagonals that bisect each other, but they are not perpendicular unless it is also a rhombus. Perpendicularity is the extra condition.",
            "A rectangle has diagonals that bisect each other AND are congruent — but they are not perpendicular unless it is a square. This is the mirror image of the rhombus.",
            "Correct. A rhombus has four equal sides, and its diagonals are perpendicular bisectors of each other. They are generally different lengths: d₁ ≠ d₂ unless the rhombus is a square. That is exactly what the question's \"not necessarily congruent\" is testing.",
            "An isosceles trapezoid has congruent diagonals, but they do not bisect each other at all — only one pair of its sides is parallel."
          ],
          "hint": "Sort the four by what their diagonals do: bisect, be congruent, be perpendicular. You want the one with bisect and perpendicular but not congruent.",
          "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,18 200,72 120,126 40,72\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"18\" x2=\"120\" y2=\"126\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><line x1=\"40\" y1=\"72\" x2=\"200\" y2=\"72\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><path d=\"M 120,60 L 132,60 L 132,72\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"126\" y=\"46\" 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\">d₁</text><text x=\"160\" y=\"66\" 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\">d₂</text><text x=\"64\" y=\"146\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">d₁ ≠ d₂ in general</text></svg>",
          "diagramCaption": "A rhombus: diagonals perpendicular, bisecting, unequal. Not drawn to scale"
        },
        {
          "id": "pqa-003",
          "topic": "Area of Quadrilaterals",
          "difficulty": "medium",
          "stem": "A rhombus has diagonals of length 10 cm and 24 cm. What is its area?",
          "choices": [
            "240 cm²",
            "120 cm²",
            "60 cm²",
            "34 cm²"
          ],
          "correctIndex": 1,
          "explanations": [
            "240 cm² multiplies the diagonals without halving. That product is the area of the RECTANGLE the rhombus is inscribed in — exactly twice too big.",
            "Correct. A rhombus has area A = ½d₁d₂ = ½(10)(24) = 120 cm². The reason: the perpendicular diagonals cut it into four right triangles, each with legs d₁/2 and d₂/2 and area ½(d₁/2)(d₂/2) = d₁d₂/8. Four of those give 4(d₁d₂/8) = ½d₁d₂.",
            "60 cm² quarters the product instead of halving it — the area of two of the four right triangles rather than all four.",
            "34 cm² adds the diagonals. Adding lengths gives a length, not an area; the units alone rule this out."
          ],
          "hint": "The diagonals cut the rhombus into four congruent right triangles. Find one triangle's area and multiply.",
          "diagram": "<svg viewBox=\"0 0 240 142\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,16 208,72 120,128 32,72\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"16\" x2=\"120\" y2=\"128\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><line x1=\"32\" y1=\"72\" x2=\"208\" y2=\"72\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><path d=\"M 120,60 L 132,60 L 132,72\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><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\">10</text><text x=\"160\" y=\"66\" 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\">24</text></svg>",
          "diagramCaption": "Diagonals 10 cm and 24 cm. Not drawn to scale"
        },
        {
          "id": "pqa-004",
          "topic": "Area of Quadrilaterals",
          "difficulty": "easy",
          "stem": "A parallelogram has a base of 14 cm and a slant side of 9 cm. Its perpendicular height is 6 cm. What is its area?",
          "choices": [
            "126 cm²",
            "84 cm²",
            "54 cm²",
            "46 cm²"
          ],
          "correctIndex": 1,
          "explanations": [
            "126 cm² uses the slant side (14 × 9) instead of the height. The slant side is longer than the perpendicular height, so this always overestimates.",
            "Correct. A parallelogram's area is A = bh = 14 × 6 = 84 cm², using the PERPENDICULAR height. Cutting the right triangle off one end and translating it to the other turns the parallelogram into a 14 × 6 rectangle of exactly the same area — which is where the formula comes from.",
            "54 cm² is 9 × 6, pairing the slant side with the height. The base and the height must be perpendicular to each other, and the base here is 14.",
            "46 cm² is the perimeter, 2(14 + 9). Perimeter is a length; area needs a product of two perpendicular lengths."
          ],
          "hint": "Slide the triangle from one end of the parallelogram to the other. What shape do you get, and which two lengths are its sides?",
          "diagram": "<svg viewBox=\"0 0 240 138\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"40,110 200,110 215,40 55,40\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"55\" y1=\"40\" x2=\"55\" y2=\"110\" stroke=\"#1f2024\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><path d=\"M 55,98 L 67,98 L 67,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"60\" 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\">6</text><text x=\"110\" y=\"128\" 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\">14</text><text x=\"28\" y=\"76\" 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\">9</text></svg>",
          "diagramCaption": "Height is measured perpendicular to the base. Not drawn to scale"
        },
        {
          "id": "pqa-005",
          "topic": "Area of Quadrilaterals",
          "difficulty": "hard",
          "stem": "A parallelogram has sides of length a and b with an included angle y. Which expression gives its area?",
          "choices": [
            "ab",
            "½ab·sin(y)",
            "ab·sin(y)",
            "ab·cos(y)"
          ],
          "correctIndex": 2,
          "explanations": [
            "ab is the area only when y = 90°, making the parallelogram a rectangle. For any other angle the figure leans over and its area is strictly less.",
            "½ab·sin(y) is the area of the TRIANGLE formed by the two sides and the included angle. A parallelogram is two such triangles, so this is exactly half the answer.",
            "Correct. The height on base b is h = a·sin(y), so A = bh = ab·sin(y). This also handles the obtuse case correctly without a separate rule, because sin(180° − y) = sin(y) — the height is the same whether you measure the angle or its supplement.",
            "ab·cos(y) uses the wrong projection. Cosine gives the component ALONG the base, which is the horizontal offset, not the perpendicular height. It would also make the area zero at y = 90°, when it should be greatest."
          ],
          "hint": "Drop the height from one vertex. Which trig ratio turns the slant side a and the angle y into that perpendicular height?"
        },
        {
          "id": "pqa-006",
          "topic": "Polygon Angle Sums",
          "difficulty": "easy",
          "stem": "What is the sum of the interior angles of a convex polygon with 15 sides?",
          "choices": [
            "2340°",
            "2700°",
            "360°",
            "156°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The interior angle sum is (n − 2)·180°. For n = 15 that is 13 × 180° = 2340°. The −2 is there because a polygon with n sides splits into n − 2 triangles when you draw all the diagonals from a single vertex.",
            "2700° is 15 × 180°, skipping the −2. Drawing diagonals from one vertex of a 15-gon produces 13 triangles, not 15.",
            "360° is the sum of the EXTERIOR angles, which is 360° for every convex polygon regardless of n. The question asks about interior angles.",
            "156° is one interior angle of a REGULAR 15-gon, 2340°/15. The question asks for the total, and it does not say the polygon is regular."
          ],
          "hint": "Fan the polygon into triangles from one vertex. How many triangles does a 15-gon give, and what does each contribute?"
        },
        {
          "id": "pqa-007",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "Each exterior angle of a regular polygon measures 24°. How many sides does the polygon have?",
          "choices": [
            "15",
            "12",
            "24",
            "156"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The exterior angles of any convex polygon total 360°, and in a regular polygon they are all equal, so n = 360°/24° = 15.",
            "12 would correspond to exterior angles of 30°. Check the division: 360/24 = 15, not 12.",
            "24 repeats the given angle measure. The number of sides is what 360° divides into, not the angle itself.",
            "156 is the INTERIOR angle, 180° − 24°. That is a real quantity here, but it is a number of degrees, not a count of sides."
          ],
          "hint": "Walk once around the polygon and you turn through a full revolution. Split that turn among the equal corners."
        },
        {
          "id": "pqa-008",
          "topic": "Quadrilateral Properties",
          "difficulty": "medium",
          "stem": "A quadrilateral has two distinct pairs of adjacent congruent sides and perpendicular diagonals, but only one diagonal is bisected. What is it?",
          "choices": [
            "Rhombus",
            "Kite",
            "Isosceles trapezoid",
            "Rectangle"
          ],
          "correctIndex": 1,
          "explanations": [
            "A rhombus has all FOUR sides congruent, and BOTH of its diagonals are bisected. The question specifies two distinct pairs and only one bisected diagonal, which rules it out.",
            "Correct. A kite is defined by two distinct pairs of adjacent congruent sides. Its diagonals are perpendicular, but only the diagonal joining the two vertices between unequal sides — the axis of symmetry — bisects the other. A rhombus is the special case where both pairs happen to be equal to each other.",
            "An isosceles trapezoid has one pair of PARALLEL sides and congruent legs, and its diagonals are congruent rather than perpendicular.",
            "A rectangle has congruent diagonals that bisect each other, and they are perpendicular only when it is a square. Its congruent sides come in OPPOSITE pairs, not adjacent ones."
          ],
          "hint": "\"Adjacent congruent sides\" is the giveaway — parallelograms pair up opposite sides instead.",
          "diagram": "<svg viewBox=\"0 0 240 156\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,14 190,80 120,142 50,80\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"14\" x2=\"120\" y2=\"142\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><line x1=\"50\" y1=\"80\" x2=\"190\" y2=\"80\" stroke=\"#1f2024\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><path d=\"M 120,68 L 132,68 L 132,80\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"146\" y=\"40\" fill=\"#6b6b76\" font-size=\"11\" 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=\"84\" y=\"40\" fill=\"#6b6b76\" font-size=\"11\" 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=\"154\" y=\"118\" fill=\"#6b6b76\" font-size=\"11\" 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=\"76\" y=\"118\" fill=\"#6b6b76\" font-size=\"11\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">b</text></svg>",
          "diagramCaption": "Two distinct pairs of adjacent equal sides. Not drawn to scale"
        },
        {
          "id": "pqa-009",
          "topic": "Quadrilateral Properties",
          "difficulty": "hard",
          "stem": "A parallelogram is known to have congruent diagonals. What can you conclude?",
          "choices": [
            "It is a rhombus.",
            "It is a rectangle.",
            "It is a square.",
            "Nothing beyond it being a parallelogram."
          ],
          "correctIndex": 1,
          "explanations": [
            "Congruent diagonals do not give a rhombus — a rhombus is what you get from PERPENDICULAR diagonals. The two conditions produce the two different specializations of a parallelogram.",
            "Correct. A parallelogram with congruent diagonals is a rectangle. The diagonals of a parallelogram already bisect each other; making them congruent as well forces all four half-diagonals to be equal, which puts the four vertices on a common circle and makes every angle a right angle.",
            "A square needs both extra conditions at once: congruent diagonals AND perpendicular ones. Only one is given here, so the figure may be a non-square rectangle.",
            "There is definitely more to conclude. Congruent diagonals are one of the standard sufficient conditions for a rectangle, so the extra information is not idle."
          ],
          "hint": "Follow the property chain: parallelogram plus perpendicular diagonals gives one figure, parallelogram plus congruent diagonals gives the other."
        },
        {
          "id": "pqa-010",
          "topic": "Area of Quadrilaterals",
          "difficulty": "medium",
          "stem": "A trapezoid has parallel sides of 8 cm and 14 cm and a height of 5 cm. What is its area?",
          "choices": [
            "110 cm²",
            "55 cm²",
            "27 cm²",
            "280 cm²"
          ],
          "correctIndex": 1,
          "explanations": [
            "110 cm² is (b₁ + b₂)h without halving, 22 × 5. The formula averages the two parallel sides before multiplying by the height.",
            "Correct. A trapezoid's area is A = ½(b₁ + b₂)h = ½(8 + 14)(5) = ½(22)(5) = 55 cm². The average of the two parallel sides, 11 cm, is the width of a rectangle with the same area and height.",
            "27 cm² adds all three numbers, 8 + 14 + 5. That mixes two parallel sides with a perpendicular height and produces neither an area nor a perimeter.",
            "280 cm² multiplies all three, 8 × 14 × 5. Three lengths multiplied give a volume, not an area — the units give it away."
          ],
          "hint": "Replace the trapezoid with a rectangle of the same height whose width is the average of the two parallel sides.",
          "diagram": "<svg viewBox=\"0 0 240 138\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"30,110 210,110 170,42 76,42\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"76\" y1=\"42\" x2=\"76\" y2=\"110\" stroke=\"#1f2024\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><path d=\"M 76,98 L 88,98 L 88,110\" fill=\"none\" stroke=\"#6b6b76\" stroke-width=\"1.4\"/><text x=\"82\" 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\">5</text><text x=\"112\" y=\"36\" 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\">8</text><text x=\"112\" y=\"128\" 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\">14</text></svg>",
          "diagramCaption": "The two parallel sides measure 8 cm and 14 cm. Not drawn to scale"
        },
        {
          "id": "pqa-011",
          "topic": "Regular Polygons",
          "difficulty": "hard",
          "stem": "A regular polygon with perimeter 60 cm has an apothem of 8 cm. What is its area?",
          "choices": [
            "480 cm²",
            "240 cm²",
            "68 cm²",
            "120 cm²"
          ],
          "correctIndex": 1,
          "explanations": [
            "480 cm² is the full product, apothem × perimeter. That double-counts: it is the area of all the triangles taken with the full side length as base rather than accounting for the ½ in each triangle's area.",
            "Correct. A regular polygon's area is A = ½(apothem)(perimeter) = ½(8)(60) = 240 cm². Splitting the polygon into triangles from the centre, each has base equal to a side and height equal to the apothem, so the total is ½ × apothem × (sum of all sides).",
            "68 cm² adds the apothem and perimeter. Adding two lengths cannot give an area.",
            "120 cm² is half the perimeter alone, or a slip that drops the apothem. Both quantities are needed."
          ],
          "hint": "Cut the polygon into triangles from the centre. Each has the apothem as its height — now add up all their bases at once."
        },
        {
          "id": "pqa-012",
          "topic": "Polygon Angle Sums",
          "difficulty": "medium",
          "stem": "A convex polygon has interior angles summing to 1440°. If it is regular, what is the measure of each exterior angle?",
          "choices": [
            "36°",
            "144°",
            "45°",
            "40°"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. From (n − 2)180° = 1440° we get n − 2 = 8, so n = 10 — a decagon. Its exterior angles total 360° and are all equal, so each is 360°/10 = 36°.",
            "144° is each INTERIOR angle, 1440°/10. The exterior angle is its supplement, 180° − 144° = 36°.",
            "45° would correspond to an octagon (n = 8). Solving (n − 2)180 = 1440 gives n = 10, not 8 — the −2 is easy to drop here.",
            "40° would correspond to a nonagon (n = 9). Recheck the division: 1440/180 = 8, so n − 2 = 8 and n = 10."
          ],
          "hint": "Recover n from the interior sum first, then use the fact that exterior angles always total 360° whatever n turns out to be."
        },
        {
          "id": "pqa-013",
          "topic": "Interior Angle Sum",
          "difficulty": "easy",
          "stem": "Find the sum of the interior angles of a decagon (n = 10).",
          "choices": [
            "1260°",
            "1440°",
            "1800°",
            "360°"
          ],
          "correctIndex": 1,
          "explanations": [
            "1260° is the interior sum for a NONAGON (n = 9): (9 − 2)·180. Off-by-one on n is the most frequent slip in this formula.",
            "Correct. The interior angle sum of an n-gon is (n − 2)·180°. For n = 10 that is (10 − 2)·180 = 8 × 180 = 1440°. The (n − 2) counts the triangles the polygon splits into from one vertex.",
            "1800° is 10 × 180, which forgets to subtract 2 from n. That would be the sum if the polygon split into ten triangles rather than eight.",
            "360° is the sum of the EXTERIOR angles, which is 360° for every convex polygon regardless of n. Interior and exterior sums behave very differently: one grows with n, the other never changes."
          ],
          "hint": "Picture drawing every diagonal from a single vertex. How many triangles does an n-gon break into?"
        },
        {
          "id": "pqa-014",
          "topic": "Regular Polygons",
          "difficulty": "medium",
          "stem": "Each interior angle of a regular polygon measures 150°. How many sides does it have?",
          "choices": [
            "10",
            "12",
            "15",
            "24"
          ],
          "correctIndex": 1,
          "explanations": [
            "10 sides would give an exterior angle of 36° and an interior angle of 144°, not 150°. Close, but the arithmetic does not land.",
            "Correct. The fastest route goes through the exterior angle: interior 150° means exterior 180 − 150 = 30°. Since exterior angles of a regular n-gon each equal 360/n, we get n = 360/30 = 12.",
            "15 sides gives an exterior angle of 24° and an interior angle of 156°. This comes from dividing 360 by the interior angle's complement rather than its supplement.",
            "24 sides gives an exterior angle of 15° — the digits of 150 misread. The exterior angle here is 30°, not 15°."
          ],
          "hint": "Convert the interior angle to its exterior partner first. The exterior route avoids solving a fraction equation."
        },
        {
          "id": "pqa-015",
          "topic": "Exterior Angles",
          "difficulty": "easy",
          "stem": "One exterior angle of a regular polygon measures 24°. How many sides does the polygon have?",
          "choices": [
            "12",
            "15",
            "18",
            "24"
          ],
          "correctIndex": 1,
          "explanations": [
            "12 sides gives an exterior angle of 30°, not 24°. This comes from dividing 360 by 30 rather than by the given 24.",
            "Correct. The exterior angles of any convex polygon sum to 360°, and in a regular polygon they are all equal. So n = 360/24 = 15.",
            "18 sides gives an exterior angle of 20°. The division was performed with the wrong divisor.",
            "24 repeats the given angle as though it were the side count. The angle and the number of sides are related by division into 360, not by equality."
          ],
          "hint": "The exterior angles always total the same amount, no matter how many sides. Divide by the size of one."
        },
        {
          "id": "pqa-016",
          "topic": "Parallelogram Angles",
          "difficulty": "medium",
          "stem": "Consecutive angles of a parallelogram measure (2x + 10)° and (3x − 40)°. Find x.",
          "choices": [
            "42",
            "50",
            "30",
            "38"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Consecutive angles of a parallelogram are SUPPLEMENTARY. So (2x + 10) + (3x − 40) = 180, giving 5x − 30 = 180, then 5x = 210 and x = 42. The angles are 94° and 86°, which do sum to 180°.",
            "50 comes from setting the two expressions equal: 2x + 10 = 3x − 40 gives x = 50. That would be right for OPPOSITE angles, which are equal — but consecutive angles are supplementary.",
            "30 comes from setting the sum to 90° instead of 180°. Consecutive angles of a parallelogram straddle a pair of parallel sides cut by a transversal, so they are co-interior and sum to a straight angle.",
            "38 comes from solving 5x = 190, which mishandles the −30 constant. Moving −30 to the right side adds 30 to 180, giving 210."
          ],
          "hint": "Decide first whether consecutive angles in a parallelogram are equal or supplementary. The two lead to very different equations."
        },
        {
          "id": "pqa-017",
          "topic": "Rhombus Diagonals",
          "difficulty": "hard",
          "stem": "A rhombus has a side of length 10 and one diagonal of length 12. Find the other diagonal.",
          "choices": [
            "8",
            "16",
            "20",
            "6"
          ],
          "correctIndex": 1,
          "explanations": [
            "8 is HALF the other diagonal, not the whole thing. The right-triangle computation gives the half-diagonal; it still has to be doubled because the diagonals bisect each other.",
            "Correct. The diagonals bisect each other perpendicularly, so half of the given diagonal is 6, and the side of 10 is the hypotenuse of a right triangle with legs 6 and h. Then h = √(100 − 36) = 8, and the full second diagonal is 2 × 8 = 16.",
            "20 doubles the side length rather than the computed half-diagonal. The side is a hypotenuse in this configuration, not a piece of a diagonal.",
            "6 is half of the GIVEN diagonal, which was an intermediate value in the calculation rather than the answer."
          ],
          "hint": "Half of each diagonal forms the legs of a right triangle whose hypotenuse is the side. Solve for the leg, then remember to double."
        },
        {
          "id": "pqa-018",
          "topic": "Reasoning: Classification",
          "difficulty": "medium",
          "stem": "A quadrilateral has diagonals that bisect each other. What is the strongest conclusion, and why is nothing stronger justified?",
          "choices": [
            "It is a rectangle, because bisecting diagonals force right angles.",
            "It is a parallelogram — bisecting diagonals guarantee that and no more.",
            "It is a rhombus, because bisecting diagonals are perpendicular.",
            "It is a square, since only squares have bisecting diagonals."
          ],
          "correctIndex": 1,
          "explanations": [
            "Rectangles need CONGRUENT diagonals, which is a separate condition. Bisecting alone does not give right angles.",
            "Correct. Diagonals bisecting each other is exactly the condition for a parallelogram. Rectangles add congruent diagonals, rhombuses add perpendicular ones, and squares add both.",
            "Rhombuses need PERPENDICULAR diagonals. Bisecting and perpendicular are different properties.",
            "Every parallelogram has bisecting diagonals, and most are not squares."
          ],
          "hint": "Each extra property on the quadrilateral family tree buys exactly one extra guarantee.",
          "kind": "reasoning"
        },
        {
          "id": "pqa-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "hard",
          "stem": "A student computes the sum of the interior angles of a regular decagon as 10 × 180° = 1800°, then says each interior angle is 180°. What went wrong?",
          "choices": [
            "The interior angle sum is (n − 2)180° = 1440°, so each angle is 144°.",
            "The sum is right, but they should have divided by 20.",
            "The interior angle sum of any polygon is 360°.",
            "Nothing is wrong; a decagon's angles are each 180°."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The sum is (n − 2)180° = 8 × 180° = 1440°, and dividing by 10 gives 144° per angle.",
            "Dividing by 20 would give 90°, which is not right either — the error is in the sum, not the division.",
            "360° is the sum of the EXTERIOR angles, which is the same for every polygon. Interior angle sums grow with n.",
            "An interior angle of 180° would be a straight line, which cannot be a corner of a polygon — the answer gives itself away."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}