{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Congruence, Similarity & Symmetry",
    "author": "Staples Education",
    "description": "Congruence criteria including why SSA fails, CPCTC, similarity and area scaling — transcribed from a worked tutoring whiteboard. Symmetry and congruent-figures questions are authored to complete the shelf; the archive does not cover them.",
    "created": "2026-08-06T04:14:56.051Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "congruence-similarity",
      "title": "Congruence, Similarity & Symmetry",
      "blurb": "The five congruence criteria and why SSA is not one of them, CPCTC, the similarity criteria and how area scales, plus lines and orders of symmetry.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "congruent-triangles",
        "similar-figures",
        "congruent-figures-review",
        "symmetry"
      ],
      "questions": [
        {
          "id": "cgs-001",
          "topic": "Congruence Criteria",
          "difficulty": "medium",
          "stem": "Which triangle congruence shortcut is INVALID, because it can produce two different triangles from the same data?",
          "choices": [
            "SSA",
            "HL",
            "AAS",
            "SAS"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Side-Side-Angle does not fix a triangle. With two sides and an angle NOT between them, the third vertex can often be placed two different ways — the swinging side reaches the base line at two points. This is the ambiguous case, and it is why SSA is not a congruence criterion.",
            "HL is valid, but only for right triangles: a congruent hypotenuse and one congruent leg force the third side by the Pythagorean theorem, so it is really SSS in disguise.",
            "AAS is valid. Two angles fix the third (they sum to 180°), so AAS becomes ASA once you name the third angle.",
            "SAS is valid. The angle sits BETWEEN the two sides, which locks the third vertex in exactly one place — the difference between SAS and SSA is entirely about where the angle is."
          ],
          "hint": "Try to build a triangle from two sides and a non-included angle. Can the free side swing to meet the base in more than one spot?",
          "diagram": "<svg viewBox=\"0 0 240 138\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"105\" x2=\"215\" y2=\"105\" stroke=\"#6b6b76\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"105\" x2=\"130\" y2=\"30\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"130\" y1=\"30\" x2=\"90\" y2=\"105\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><line x1=\"130\" y1=\"30\" x2=\"185\" y2=\"105\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><circle cx=\"25\" cy=\"105\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"130\" cy=\"30\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"90\" cy=\"105\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"185\" cy=\"105\" r=\"3.2\" fill=\"#c0392b\"/><text x=\"38\" y=\"98\" 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\">θ</text><text x=\"66\" y=\"62\" fill=\"#7c3aed\" 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=\"108\" y=\"74\" 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\">b</text><text x=\"166\" y=\"66\" fill=\"#c0392b\" 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=\"72\" y=\"128\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">two possible triangles</text></svg>",
          "diagramCaption": "The same two sides and non-included angle close two different ways. Not drawn to scale"
        },
        {
          "id": "cgs-002",
          "topic": "Congruence Criteria",
          "difficulty": "easy",
          "stem": "What does the abbreviation CPCTC stand for in geometric proofs?",
          "choices": [
            "Corresponding Parts of Congruent Triangles are Congruent",
            "Corresponding Parts of Central Triangles are Congruent",
            "Calculated Proportions of Congruent Triangles create Congruence",
            "Congruent Parts of Congruent Triangles are Congruent"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Once two triangles are proved congruent, every one of the six corresponding parts — three sides and three angles — is congruent. It is the standard closing move of a two-stage proof: establish congruence by a criterion, then quote CPCTC to collect the specific part you wanted.",
            "\"Central\" has no meaning here. The word is \"Corresponding\", which names the matching-up of vertices that the congruence statement declares.",
            "Nothing is calculated and no proportions are involved — proportions belong to SIMILAR triangles, not congruent ones.",
            "This says congruent parts are congruent, which is circular and useless. The content of CPCTC is that CORRESPONDING parts become congruent once the triangles are."
          ],
          "hint": "Two of these four differ by one word. Which word describes how the vertices are matched up?"
        },
        {
          "id": "cgs-003",
          "topic": "Similar Figures",
          "difficulty": "medium",
          "stem": "Two similar triangles have a side-length scale factor of 2 : 5. What is the ratio of their areas?",
          "choices": [
            "2 : 5",
            "4 : 25",
            "8 : 125",
            "4 : 10"
          ],
          "correctIndex": 1,
          "explanations": [
            "2 : 5 is the ratio of LENGTHS. Areas grow faster, because an area is a product of two lengths.",
            "Correct. The ratio of areas is the square of the ratio of lengths: 2² : 5² = 4 : 25.",
            "8 : 125 is the cube, which is the ratio of VOLUMES for similar solids. This question is about a flat figure.",
            "4 : 10 doubles both parts, which reduces straight back to 2 : 5 — the length ratio again."
          ],
          "hint": "Length ratio, area ratio, volume ratio: first power, second power, third power."
        },
        {
          "id": "cgs-004",
          "topic": "Similarity Criteria",
          "difficulty": "medium",
          "stem": "Which condition is enough on its own to prove two triangles SIMILAR?",
          "choices": [
            "Two pairs of congruent angles (AA)",
            "Two pairs of congruent sides (SS)",
            "Three pairs of congruent angles AND one pair of congruent sides",
            "Any pair of congruent angles"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. AA is enough. If two angles match, the third must match too (all three sum to 180°), and equal angles force the sides into proportion. Similarity needs only shape, not size — which is exactly why it takes less evidence than congruence.",
            "Two pairs of congruent sides says nothing without knowing the angle between them or the third side. The similarity version is SAS~, which needs two pairs of PROPORTIONAL sides and the included angle.",
            "This is more than enough — in fact a congruent side alongside three congruent angles proves the triangles CONGRUENT, which is stronger than similar. The question asks for what suffices, and AA already does.",
            "One pair of congruent angles is not enough. Two triangles can share a 40° angle and be nothing alike."
          ],
          "hint": "How many angles do you need before the third is determined for free?"
        },
        {
          "id": "cgs-005",
          "topic": "Congruence Criteria",
          "difficulty": "medium",
          "stem": "Two right triangles have congruent hypotenuses and one pair of congruent legs. Which criterion proves them congruent?",
          "choices": [
            "HL",
            "SSA",
            "AAA",
            "They cannot be proved congruent"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Hypotenuse-Leg applies only to right triangles, and it works because the right angle plus the Pythagorean theorem force the remaining leg: if the hypotenuse and one leg match, the other leg must too. HL is SSS with the third side computed rather than measured.",
            "The data here IS side-side-angle in form — two sides and a non-included right angle — which is normally ambiguous. The right angle is the special case that removes the ambiguity, and HL is the name for that case.",
            "AAA proves similarity, never congruence. Three matching angles fix the shape but say nothing about size.",
            "They certainly can. HL is one of the five standard criteria precisely for this configuration."
          ],
          "hint": "A right triangle's third side is never independent. What determines it once you know the other two?"
        },
        {
          "id": "cgs-006",
          "topic": "Congruent Figures",
          "difficulty": "easy",
          "stem": "Figure B was produced from figure A by a translation followed by a reflection. What is the relationship between A and B?",
          "choices": [
            "Congruent",
            "Similar but not congruent",
            "Neither similar nor congruent",
            "Congruent only if the reflection was over an axis"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Translations and reflections are both rigid motions — they preserve every distance and angle — and composing rigid motions gives another rigid motion. So B is congruent to A. Its orientation is reversed by the reflection, but reversed orientation does not affect congruence.",
            "Similar-but-not-congruent requires a size change, which only a dilation with k ≠ 1 produces. Neither transformation here changes size.",
            "Every rigid motion preserves both shape and size, so the figures cannot fail to be at least congruent.",
            "The line of reflection makes no difference. Reflecting over any line — axis, diagonal, or otherwise — preserves distances."
          ],
          "hint": "Sort the four transformations into those that change size and those that do not. Which kind were both of these?"
        },
        {
          "id": "cgs-007",
          "topic": "Congruent Figures",
          "difficulty": "medium",
          "stem": "Triangle ABC maps onto triangle DEF under a sequence of rigid motions, with A → D, B → E, C → F. Which statement must be true?",
          "choices": [
            "∠A ≅ ∠D and AB ≅ DE",
            "∠A ≅ ∠E and AB ≅ EF",
            "AB ≅ DF only",
            "The triangles have equal perimeters but possibly different angles"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The correspondence stated by the mapping is what tells you which parts match: A pairs with D, so ∠A ≅ ∠D, and side AB pairs with side DE. Reading the correspondence off the statement is the whole skill here.",
            "This pairs A with E, contradicting the given mapping. A congruence statement is an ordered claim — the order of the letters IS the correspondence.",
            "AB corresponds to DE, not DF. And it is not the only congruent pair: all six corresponding parts are congruent.",
            "Rigid motions preserve angles as well as lengths, so the angles match too. Equal perimeters with different angles would describe two figures that are not congruent at all."
          ],
          "hint": "Read the letters in the order given. That order is not decoration — it names which part matches which."
        },
        {
          "id": "cgs-008",
          "topic": "Symmetry",
          "difficulty": "easy",
          "stem": "How many lines of symmetry does a regular hexagon have?",
          "choices": [
            "3",
            "6",
            "12",
            "0"
          ],
          "correctIndex": 1,
          "explanations": [
            "3 counts only the lines through opposite vertices. There are three more through the midpoints of opposite sides.",
            "Correct. A regular n-gon has exactly n lines of symmetry, so a regular hexagon has 6: three joining opposite vertices, and three joining the midpoints of opposite sides.",
            "12 is the size of the hexagon's full symmetry group — 6 reflections plus 6 rotations. The question asks only for the reflection lines.",
            "0 would describe a figure with no mirror symmetry at all, like a general scalene triangle. A regular hexagon is as symmetric as a six-sided figure gets."
          ],
          "hint": "Two kinds of mirror line are available: through opposite corners, and through opposite edge midpoints. Count both.",
          "diagram": "<svg viewBox=\"0 0 240 162\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,15 178,48 178,114 120,147 62,114 62,48\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"120\" y1=\"15\" x2=\"120\" y2=\"147\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"178\" y1=\"48\" x2=\"62\" y2=\"114\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"178\" y1=\"114\" x2=\"62\" y2=\"48\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"90\" y1=\"31\" x2=\"150\" y2=\"131\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"150\" y1=\"31\" x2=\"90\" y2=\"131\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><line x1=\"62\" y1=\"81\" x2=\"178\" y2=\"81\" stroke=\"#6b6b76\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/></svg>",
          "diagramCaption": "Three through opposite vertices, three through opposite side midpoints. Not drawn to scale"
        },
        {
          "id": "cgs-009",
          "topic": "Symmetry",
          "difficulty": "medium",
          "stem": "A parallelogram that is not a rectangle or a rhombus has how many lines of symmetry, and what order of rotational symmetry?",
          "choices": [
            "0 lines, order 2",
            "2 lines, order 2",
            "1 line, order 1",
            "0 lines, order 1"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A general parallelogram has NO line of symmetry — folding it along either diagonal or either midline does not match the halves up. But it does have rotational symmetry of order 2: turning it 180° about the intersection of its diagonals maps it onto itself.",
            "2 lines would describe a rectangle or a rhombus. Adding right angles or equal sides is exactly what buys the mirror lines.",
            "1 line describes an isosceles trapezoid or a kite. A parallelogram has none.",
            "Order 1 means no rotational symmetry beyond a full turn. A parallelogram genuinely does map onto itself at 180°, which is order 2."
          ],
          "hint": "Try folding a leaning parallelogram in half — any way you like. Then try spinning it half a turn."
        },
        {
          "id": "cgs-010",
          "topic": "Symmetry",
          "difficulty": "medium",
          "stem": "A figure maps onto itself after a rotation of 72° about its centre. What is the smallest number of sides it could have as a regular polygon?",
          "choices": [
            "4",
            "5",
            "6",
            "72"
          ],
          "correctIndex": 1,
          "explanations": [
            "A square maps onto itself every 90°, not every 72°.",
            "Correct. A regular n-gon maps onto itself every 360°/n. Setting 360/n = 72 gives n = 5 — a regular pentagon.",
            "A regular hexagon has a 60° rotational symmetry, not 72°.",
            "72 is the angle in degrees, not a count of sides. A regular 72-gon would map onto itself every 5°."
          ],
          "hint": "The smallest rotation that works for a regular n-gon is one nth of a full turn."
        },
        {
          "id": "cgs-011",
          "topic": "Similar Figures",
          "difficulty": "hard",
          "stem": "In △ABC, a line parallel to BC cuts AB at D and AC at E. If AD = 4, DB = 6 and AE = 6, what is EC?",
          "choices": [
            "4",
            "9",
            "10",
            "6"
          ],
          "correctIndex": 1,
          "explanations": [
            "4 repeats AD. The two sides are cut in the same RATIO, not to the same lengths.",
            "Correct. A line parallel to one side cuts the other two proportionally (the Side-Splitter Theorem, which follows from △ADE ~ △ABC by AA). So AD/DB = AE/EC gives 4/6 = 6/EC, and EC = 36/4 = 9.",
            "10 would make AC = 16 and AE/EC = 6/10 = 3/5, which is not 4/6 = 2/3.",
            "6 repeats AE. It would make the split 6:6 on one side and 4:6 on the other — not proportional."
          ],
          "hint": "The parallel line creates a smaller triangle similar to the whole. Set the two side splits into a proportion.",
          "diagram": "<svg viewBox=\"0 0 240 148\" width=\"100%\" style=\"max-width:240px; height:auto; display:block; margin:0 auto;\" role=\"img\"><polygon points=\"120,20 30,125 210,125\" fill=\"none\" stroke=\"#7c3aed\" stroke-width=\"2\" stroke-linejoin=\"round\"/><line x1=\"72\" y1=\"76\" x2=\"168\" y2=\"76\" stroke=\"#1f2024\" stroke-width=\"2\" stroke-linecap=\"round\"/><circle cx=\"72\" cy=\"76\" r=\"3.2\" fill=\"#1f2024\"/><circle cx=\"168\" cy=\"76\" r=\"3.2\" fill=\"#1f2024\"/><text x=\"112\" y=\"14\" 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=\"20\" y=\"138\" 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=\"212\" y=\"138\" 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\">C</text><text x=\"56\" y=\"74\" 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\">D</text><text x=\"174\" y=\"74\" 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\">E</text><text x=\"86\" 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\">4</text><text x=\"50\" y=\"106\" 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\">6</text><text x=\"150\" 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\">6</text><text x=\"186\" y=\"106\" fill=\"#c0392b\" font-size=\"12\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">?</text></svg>",
          "diagramCaption": "DE ∥ BC. Not drawn to scale"
        },
        {
          "id": "cgs-012",
          "topic": "Similar Figures",
          "difficulty": "medium",
          "stem": "Two regular pentagons have perimeters of 30 cm and 45 cm. What is the ratio of their areas?",
          "choices": [
            "2 : 3",
            "4 : 9",
            "8 : 27",
            "30 : 45"
          ],
          "correctIndex": 1,
          "explanations": [
            "2 : 3 is the ratio of perimeters (and so of side lengths), reduced. Areas scale by the square of that.",
            "Correct. All regular pentagons are similar, so the perimeter ratio 30 : 45 = 2 : 3 is the length ratio, and the area ratio is 2² : 3² = 4 : 9.",
            "8 : 27 is the cube, which would be the volume ratio for similar solids. These are flat figures.",
            "30 : 45 is the unreduced perimeter ratio, which is the LENGTH ratio, not the area ratio."
          ],
          "hint": "Perimeter is a length, so the perimeter ratio is the length ratio. Now square it."
        },
        {
          "id": "cgs-013",
          "topic": "Invalid Congruence Criteria",
          "difficulty": "medium",
          "stem": "Which of the following is NOT a valid triangle congruence criterion?",
          "choices": [
            "SAS",
            "ASA",
            "SSA",
            "HL"
          ],
          "correctIndex": 2,
          "explanations": [
            "SAS is valid: two sides and the angle BETWEEN them fix the triangle completely, because the included angle locks the two sides into one configuration.",
            "ASA is valid: two angles and the included side determine the third angle by the angle sum, and from there the whole triangle.",
            "Correct. SSA is not a valid criterion. With two sides and a non-included angle, the third side can sometimes swing to two different positions, producing two non-congruent triangles — the ambiguous case that returns in the Law of Sines.",
            "HL is valid, but only for RIGHT triangles: hypotenuse and one leg determine the other leg by the Pythagorean Theorem, which reduces it to SSS."
          ],
          "hint": "One of these leaves room for two different triangles from the same data. That ambiguity is what disqualifies it."
        },
        {
          "id": "cgs-014",
          "topic": "Midsegment Theorem",
          "difficulty": "easy",
          "stem": "A midsegment of a triangle measures 7 cm. How long is the side it is parallel to?",
          "choices": [
            "3.5 cm",
            "7 cm",
            "14 cm",
            "21 cm"
          ],
          "correctIndex": 2,
          "explanations": [
            "3.5 cm halves the midsegment. The relationship runs the other way: the midsegment is half the side, so the side is the larger of the two.",
            "7 cm would make the midsegment equal to the side it parallels, which never happens in a genuine triangle.",
            "Correct. The Midsegment Theorem says a segment joining the midpoints of two sides is parallel to the third side and half its length. So the third side is 2 × 7 = 14 cm.",
            "21 cm triples the midsegment. The factor in the Midsegment Theorem is 2, not 3."
          ],
          "hint": "Decide which of the two segments the theorem describes as 'half'. That tells you whether to double or halve."
        },
        {
          "id": "cgs-015",
          "topic": "Similar Triangles",
          "difficulty": "medium",
          "stem": "△PQR ~ △XYZ with PQ = 12 and XY = 8. If the perimeter of △XYZ is 30, find the perimeter of △PQR.",
          "choices": [
            "20",
            "45",
            "50",
            "67.5"
          ],
          "correctIndex": 1,
          "explanations": [
            "20 scales in the wrong direction, multiplying by 8/12 instead of 12/8. Since PQ is longer than XY, △PQR is the larger triangle and its perimeter must exceed 30.",
            "Correct. The scale factor from XYZ to PQR is 12/8 = 1.5. Perimeter is a length measurement, so it scales by the same factor: 30 × 1.5 = 45.",
            "50 does not follow from any consistent scaling of 30. Apply the ratio directly rather than estimating.",
            "67.5 applies the scale factor twice, 30 × 1.5². Squaring the factor is correct for AREA, not for perimeter — perimeter is one-dimensional."
          ],
          "hint": "Perimeter is a sum of lengths, so it scales the same way a single length does."
        },
        {
          "id": "cgs-016",
          "topic": "AA Similarity",
          "difficulty": "easy",
          "stem": "Two triangles have two pairs of congruent angles. What can be concluded?",
          "choices": [
            "They are congruent.",
            "They are similar.",
            "Nothing — two angles is not enough.",
            "They are congruent only if a side is also known."
          ],
          "correctIndex": 1,
          "explanations": [
            "Congruence requires matching SIZE as well as shape. Two triangles can have identical angles and wildly different sizes — think of a photograph and its enlargement.",
            "Correct. AA similarity: if two angles of one triangle are congruent to two of another, the third pair matches automatically by the angle sum, so all three angles agree and the triangles are similar with proportional sides.",
            "Two angles is entirely sufficient for similarity. Because the angles of a triangle sum to 180°, knowing two determines the third at no extra cost.",
            "This is true as a statement about congruence but it is not the strongest conclusion available. Similarity already follows from the two angles alone, with no side needed."
          ],
          "hint": "Distinguish what two angles fix — shape or size — and name the relationship that captures only that."
        },
        {
          "id": "cgs-017",
          "topic": "Similarity Ratios",
          "difficulty": "hard",
          "stem": "△ABC ~ △DEF with AB = 6, DE = 9, and BC = 8. Find EF.",
          "choices": [
            "5.33",
            "11",
            "12",
            "10.5"
          ],
          "correctIndex": 2,
          "explanations": [
            "5.33 is 8 × (6/9), scaling in the wrong direction. Since DE is larger than AB, △DEF is the bigger triangle and EF must exceed BC.",
            "11 adds the difference, 8 + 3, instead of multiplying by the ratio. Similarity scales lengths multiplicatively, not by adding a constant.",
            "Correct. The scale factor from ABC to DEF is DE/AB = 9/6 = 1.5. Corresponding sides scale by that factor, so EF = BC × 1.5 = 8 × 1.5 = 12.",
            "10.5 comes from an inconsistent ratio. Verify by checking that EF/BC equals DE/AB: 12/8 = 1.5 = 9/6. ✓"
          ],
          "hint": "Establish the scale factor from the pair of sides you know completely, then check its direction before applying it."
        },
        {
          "id": "cgs-018",
          "topic": "Reasoning: Choose the Criterion",
          "difficulty": "hard",
          "stem": "Two triangles share two pairs of congruent sides and one pair of congruent angles that is NOT between those sides. Why is congruence not guaranteed?",
          "choices": [
            "This is SSA, which can describe two genuinely different triangles — the ambiguous case.",
            "It is SAS, which is valid, so congruence IS guaranteed.",
            "Two sides and an angle are never enough information.",
            "It is only invalid when the triangles are obtuse."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. With the angle outside the two sides, the third side can often be placed two different ways, producing two non-congruent triangles. That is why SSA is not a criterion.",
            "SAS requires the angle to be INCLUDED between the two sides. Here it is not, which is the whole difference.",
            "SAS is two sides and an angle and is perfectly valid — the position of the angle is what matters.",
            "The ambiguity arises from the configuration, not from whether the triangle is obtuse."
          ],
          "hint": "Where the angle sits relative to the two sides is the entire question.",
          "kind": "reasoning"
        },
        {
          "id": "cgs-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "medium",
          "stem": "Two similar triangles have a scale factor of 4. A student says the perimeter of the larger is 16 times the smaller. What is the correct factor?",
          "choices": [
            "16, since perimeter scales with the square.",
            "4 — perimeter is a sum of LENGTHS, so it scales by the linear factor.",
            "64, matching the cube.",
            "The perimeter is unchanged."
          ],
          "correctIndex": 1,
          "explanations": [
            "16 = 4² is how AREA scales. Perimeter is not an area.",
            "Correct. Perimeter is the sum of side lengths, and every length scales by 4 — so the sum does too.",
            "64 = 4³ is how volume scales, and these are two-dimensional figures.",
            "Similar figures with a scale factor other than 1 have different perimeters."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}