{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Logic Foundations",
    "author": "Staples Education",
    "description": "Category 1 of the Geometry sequence: conditional statements and their truth tables, De Morgan's Laws, negating quantified statements, converse/inverse/contrapositive, the Laws of Syllogism and Detachment, Euclid's postulates, and the role of CPCTC. Converted from worked one-to-one tutoring sessions — each distractor is an error a student actually made.",
    "created": "2026-08-06T04:14:55.932Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "logic-foundations",
      "title": "Logic Foundations",
      "blurb": "Conditionals, negation and De Morgan's Laws, quantifiers, the contrapositive, and the postulate structure every later proof leans on.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "logic"
      ],
      "questions": [
        {
          "id": "log-001",
          "topic": "Conditional Statements",
          "difficulty": "medium",
          "stem": "Evaluate the truth value of the following conditional statement: “If a triangle has four sides, then squares are circles.”",
          "choices": [
            "True",
            "False",
            "Undefined",
            "Neither — it is a biconditional"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A conditional is false in exactly one situation: a true hypothesis with a false conclusion. Here the hypothesis (“a triangle has four sides”) is false, so that situation never arises and the statement is vacuously true. Nothing about squares or circles has to be checked.",
            "False is the intuitive answer, and it comes from grading the conclusion on its own — “squares are circles” is obviously wrong, so the whole thing feels wrong. But a conditional does not assert its conclusion; it only promises the conclusion whenever the hypothesis holds. That promise is never tested here.",
            "Undefined is not one of the values a conditional can take. Every combination of hypothesis and conclusion truth values has a defined result in the truth table — that is what makes truth tables finite and checkable.",
            "A biconditional is “p if and only if q”, which asserts the implication in both directions. This statement says “if… then…” once, in one direction, so it is a plain conditional."
          ],
          "hint": "Look at the hypothesis before the conclusion. What does a conditional promise when its hypothesis is false?"
        },
        {
          "id": "log-002",
          "topic": "Negation and De Morgan",
          "difficulty": "medium",
          "stem": "By De Morgan's Laws, what is the logical equivalent of the negation ¬(p ∧ q)?",
          "choices": [
            "¬p ∨ ¬q",
            "p ∨ q",
            "¬p ∧ ¬q",
            "q → p"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Negating an “and” turns it into an “or” of the negated parts: ¬(p ∧ q) ≡ ¬p ∨ ¬q. For the pair to fail, it is enough that one of them fails — you are not required to say which.",
            "p ∨ q drops the negation entirely. This is the disjunction of the original statements, not of their denials, so it is true in cases where ¬(p ∧ q) is false.",
            "¬p ∧ ¬q negates both parts but keeps the “and”, which is the classic De Morgan slip. It claims both fail. ¬(p ∧ q) only claims that they do not both hold — a much weaker statement, satisfied when exactly one fails.",
            "q → p is a conditional, and De Morgan's Laws are about distributing a negation across ∧ and ∨. No implication appears in ¬(p ∧ q)."
          ],
          "hint": "The negation flips the connective as well as the parts. Ask what has to go wrong for “both p and q” to be untrue — one failure, or two?"
        },
        {
          "id": "log-003",
          "topic": "Quantifiers",
          "difficulty": "medium",
          "stem": "What is the correct logical negation of the universal statement “All angles are acute”?",
          "choices": [
            "At least one angle is not acute.",
            "All angles are obtuse.",
            "Some angles are obtuse.",
            "No angles are acute."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The negation of “all A are B” is “at least one A is not B”. To disprove a universal claim you only have to produce one counterexample — a single non-acute angle is enough, and it need not be obtuse (a right angle would do).",
            "“All angles are obtuse” is another universal claim, not the denial of the first one. Both statements can be false together, which a statement and its negation can never be.",
            "“Some angles are obtuse” is too specific in one direction and too weak in another. A counterexample only has to be non-acute; right angles are non-acute without being obtuse, so this misses cases where the original is genuinely false.",
            "“No angles are acute” is the strongest possible opposite — it denies every instance. The negation only needs to deny one."
          ],
          "hint": "You are trying to break a claim about everything. What is the smallest amount of evidence that does it?"
        },
        {
          "id": "log-004",
          "topic": "Converse, Inverse, Contrapositive",
          "difficulty": "medium",
          "stem": "Consider the true statement “If two angles are vertical, then they are congruent.” Which related statement must also be true?",
          "choices": [
            "If two angles are congruent, then they are vertical.",
            "If two angles are not congruent, then they are not vertical.",
            "If two angles are not vertical, then they are not congruent.",
            "Two angles are vertical if and only if they are congruent."
          ],
          "correctIndex": 1,
          "explanations": [
            "This is the converse, and it is false: two angles can measure 40° each without sitting opposite one another at an intersection. A true conditional says nothing about its converse.",
            "Correct. This is the contrapositive, ¬q → ¬p, which is logically equivalent to the original conditional — the two are true and false together, always. It is why proof by contrapositive is a valid technique rather than a trick.",
            "This is the inverse, ¬p → ¬q. Like the converse it is not equivalent to the original: plenty of non-vertical angle pairs are congruent, so it is false here.",
            "A biconditional requires the conditional AND its converse to hold. The converse fails here, so the biconditional fails with it."
          ],
          "hint": "Of the converse, inverse, and contrapositive, exactly one is always equivalent to the original. Which one negates both parts and swaps them?"
        },
        {
          "id": "log-005",
          "topic": "Conditional Statements",
          "difficulty": "easy",
          "stem": "In the conditional “If a figure is a square, then it is a rectangle”, which part is the hypothesis?",
          "choices": [
            "“it is a rectangle”",
            "“a figure is a square”",
            "The whole statement",
            "Neither — a conditional has no hypothesis"
          ],
          "correctIndex": 1,
          "explanations": [
            "“It is a rectangle” follows “then”, which makes it the conclusion. The conclusion is what the statement promises, not what triggers the promise.",
            "Correct. The hypothesis is the clause introduced by “if” — the condition that has to hold before the statement claims anything. Here that is “a figure is a square”.",
            "A conditional is built from two distinct clauses. Calling the whole statement the hypothesis loses the structure that makes it testable.",
            "Every conditional has both a hypothesis and a conclusion by definition; that is what makes it a conditional rather than a simple assertion."
          ],
          "hint": "One clause follows “if” and the other follows “then”. Which one sets the condition?"
        },
        {
          "id": "log-006",
          "topic": "Valid Inference",
          "difficulty": "medium",
          "stem": "You know “If it is raining, then the field is wet” is true, and you observe that the field is wet. What can you validly conclude?",
          "choices": [
            "It is raining.",
            "It is not raining.",
            "Nothing about the rain.",
            "It rained earlier but has stopped."
          ],
          "correctIndex": 2,
          "explanations": [
            "Concluding the hypothesis from the conclusion is affirming the consequent, and it is invalid. The conditional rules out rain-with-a-dry-field; it never claims rain is the only way a field gets wet.",
            "Nothing licenses denying the rain either. The observation is consistent with rain and with no rain, so it cannot establish either one.",
            "Correct. A true conditional plus a true conclusion tells you nothing about the hypothesis. Sprinklers, a burst pipe, or rain all leave the field wet, and the statement does not distinguish between them. (What WOULD be valid: a dry field would let you conclude it is not raining — that is the contrapositive.)",
            "This adds a detail the premises never supplied. Nothing here mentions timing at all."
          ],
          "hint": "Work out which direction of the conditional you are travelling in, and check whether that direction is the one the statement actually guarantees."
        },
        {
          "id": "log-007",
          "topic": "Postulates and Proof",
          "difficulty": "medium",
          "stem": "Which of Euclid's five postulates uniquely defines the geometry of a flat plane, guaranteeing that parallel lines never meet?",
          "choices": [
            "The Fifth Postulate (the Parallel Postulate)",
            "The Fourth Postulate",
            "The Third Postulate",
            "The First Postulate"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. The Fifth Postulate is independent of the first four: they hold on a sphere and on a saddle too, where “parallel” behaves quite differently. Adding the fifth is exactly what pins the geometry down to a flat plane — which is why dropping it produces the non-Euclidean geometries rather than a contradiction.",
            "The Fourth Postulate states that all right angles are equal to one another. It fixes a common standard for angle measure, and it holds in curved geometries as well.",
            "The Third Postulate says a circle can be drawn with any centre and radius. It is a construction rule and carries no information about parallels.",
            "The First Postulate says a straight line can be drawn between any two points. It is the most basic of the five and is satisfied in curved geometries too."
          ],
          "hint": "Four of the five survive unchanged on a curved surface. The odd one out is the one that fails there."
        },
        {
          "id": "log-008",
          "topic": "Negation and De Morgan",
          "difficulty": "hard",
          "stem": "A theorem states “If a quadrilateral is a rhombus, then its diagonals are perpendicular.” To disprove it by counterexample, what must a single figure satisfy?",
          "choices": [
            "It is a rhombus and its diagonals are perpendicular.",
            "It is a rhombus and its diagonals are not perpendicular.",
            "It is not a rhombus and its diagonals are not perpendicular.",
            "It is not a rhombus and its diagonals are perpendicular."
          ],
          "correctIndex": 1,
          "explanations": [
            "This figure satisfies the theorem rather than breaking it. An example that obeys a claim is not evidence against it, however many you collect.",
            "Correct. ¬(p → q) is equivalent to p ∧ ¬q — the one and only case a conditional forbids. A counterexample must therefore meet the hypothesis and fail the conclusion simultaneously. (No such rhombus exists, which is why the theorem stands.)",
            "Failing the hypothesis makes the conditional vacuously true for that figure, so this cannot be a counterexample. It is outside the theorem's scope entirely.",
            "A non-rhombus with perpendicular diagonals — a kite, say — is perfectly compatible with the theorem. The theorem never claimed rhombi were the only such figures; that would be its converse."
          ],
          "hint": "Write the negation of p → q as a conjunction first. A conditional is false in exactly one row of its truth table — recreate that row."
        },
        {
          "id": "log-009",
          "topic": "Valid Inference",
          "difficulty": "hard",
          "stem": "Given “If a figure is a square, then it is a rectangle” and “If a figure is a rectangle, then it has four right angles”, what does the Law of Syllogism let you conclude?",
          "choices": [
            "If a figure has four right angles, then it is a square.",
            "If a figure is a square, then it has four right angles.",
            "If a figure is a rectangle, then it is a square.",
            "Every figure with four right angles is a rectangle."
          ],
          "correctIndex": 1,
          "explanations": [
            "This reverses the chain. Syllogism carries you forward from the first hypothesis to the last conclusion, never backward — and the reversal is false anyway, since a non-square rectangle has four right angles.",
            "Correct. The Law of Syllogism chains p → q with q → r to give p → r. Here p is “is a square”, q is “is a rectangle”, and r is “has four right angles”, so the chain yields “square → four right angles”.",
            "This is the converse of the first premise, which syllogism does not license. A rectangle need not be a square.",
            "This is the converse of the second premise. True as it happens, but it does not follow from these premises by syllogism — it would need to be established separately."
          ],
          "hint": "Line the two conditionals up so the conclusion of one is the hypothesis of the other, then read off the two ends."
        },
        {
          "id": "log-010",
          "topic": "Conditional Statements",
          "difficulty": "easy",
          "stem": "A conditional statement p → q is FALSE in exactly which case?",
          "choices": [
            "p is true and q is true",
            "p is false and q is true",
            "p is true and q is false",
            "p is false and q is false"
          ],
          "correctIndex": 2,
          "explanations": [
            "Both true is the case the statement was designed to describe — the promise was made and kept, so the conditional holds.",
            "A false hypothesis makes the conditional vacuously true regardless of q. The promise was never triggered, so it cannot have been broken.",
            "Correct. A conditional is a promise: “whenever p, then q”. The only way to break it is for p to happen and q to fail. Every other row of the truth table is true.",
            "Both false is again a vacuous case — the hypothesis never fired, so nothing was promised and nothing was broken."
          ],
          "hint": "Read p → q as a promise and ask what it would take to accuse someone of breaking it."
        },
        {
          "id": "log-011",
          "topic": "Postulates and Proof",
          "difficulty": "medium",
          "stem": "In a two-stage congruence proof, what is the primary role of CPCTC (Corresponding Parts of Congruent Triangles are Congruent)?",
          "choices": [
            "To prove that two lines are parallel.",
            "To verify that the Triangle Inequality holds.",
            "To bisect an existing angle.",
            "To extract a specific pair of equal sides or angles once congruence is already established."
          ],
          "correctIndex": 3,
          "explanations": [
            "Parallelism is proved from angle relationships across a transversal. CPCTC may supply one of those angle equalities, but that is CPCTC being used, not what it is.",
            "The Triangle Inequality is a fact about side lengths in any single triangle and needs no congruence at all.",
            "Bisecting is a construction. CPCTC proves things about a figure that already exists; it does not build anything.",
            "Correct. CPCTC is not itself a criterion — it is what you harvest after a criterion has done its job. The logic of nearly every congruence proof runs in two beats: first establish △ ≅ △ by SSS, SAS, ASA, AAS or HL, then quote CPCTC to collect the one equal side or angle you actually wanted. It gives you all six correspondences for free."
          ],
          "hint": "Ask whether it is used to establish congruence or to spend it once you have it."
        },
        {
          "id": "log-012",
          "topic": "Converse, Inverse, Contrapositive",
          "difficulty": "medium",
          "stem": "The statement “If two angles form a linear pair, then they are supplementary” is true, and so is its converse-style partner “If two adjacent angles are supplementary and share a common ray, they form a linear pair.” What does this pair of facts justify writing?",
          "choices": [
            "A biconditional: angles form a linear pair if and only if they are adjacent, share a ray, and are supplementary.",
            "Only the original conditional; the converse adds nothing.",
            "The inverse of the original statement.",
            "Nothing — a statement and its converse cannot both be true."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. When a conditional and its converse are both true, the two conditions are interchangeable, and that is precisely what a biconditional (“if and only if”) records. Definitions in geometry are biconditional for this reason.",
            "The converse adds a great deal: it lets you argue in the other direction, from supplementary-and-adjacent back to linear pair. That is a second inference the original alone does not license.",
            "The inverse negates both parts of the original. It is equivalent to the converse, not to the pair of them, and it is not what having both directions buys you.",
            "A statement and its converse can absolutely both be true — that is the ordinary situation for a definition. They are simply independent of one another, so each has to be checked separately."
          ],
          "hint": "You now have the implication running in both directions. Is there a single connective that says exactly that?"
        },
        {
          "id": "log-013",
          "topic": "Contrapositive",
          "difficulty": "easy",
          "stem": "Given the statement \"If it rains, then the game is cancelled,\" which of the following is its contrapositive?",
          "choices": [
            "If the game is cancelled, then it rained.",
            "If it does not rain, then the game is not cancelled.",
            "If the game is not cancelled, then it did not rain.",
            "It rains and the game is not cancelled."
          ],
          "correctIndex": 2,
          "explanations": [
            "This is the CONVERSE — it swaps the hypothesis and conclusion without negating either. The converse is not logically equivalent to the original: the game could be cancelled for a reason other than rain, which would make this false while the original stays true.",
            "This is the INVERSE — it negates both parts but leaves them in their original order. Like the converse, it is not equivalent to the original, and for the same reason: some other cause could still cancel the game.",
            "Correct. The contrapositive negates BOTH parts AND swaps them: ¬q → ¬p. It is the only one of the three related conditionals that is logically equivalent to the original, which is why proving the contrapositive is a legitimate way to prove the original.",
            "This is the NEGATION of the conditional, not a related conditional at all. \"If p then q\" is false exactly when p is true and q is false, so this statement asserts the original is false rather than restating it."
          ],
          "hint": "Of converse, inverse and contrapositive, only one is logically equivalent to the original. Ask which operations — swapping, negating, or both — produce it."
        },
        {
          "id": "log-014",
          "topic": "Logical Equivalence",
          "difficulty": "medium",
          "stem": "A conditional statement is known to be true. Which related statement must also be true?",
          "choices": [
            "Its converse",
            "Its inverse",
            "Its contrapositive",
            "All three must be true"
          ],
          "correctIndex": 2,
          "explanations": [
            "The converse (q → p) can be false while the original is true. \"If a figure is a square, then it is a rectangle\" is true; its converse \"if a rectangle, then a square\" is false. One counterexample settles it.",
            "The inverse (¬p → ¬q) can likewise be false. \"If not a square, then not a rectangle\" fails for any non-square rectangle. The inverse is the contrapositive of the converse, so it stands or falls with the converse — not with the original.",
            "Correct. Only the contrapositive (¬q → ¬p) is logically equivalent to the original. Every row of the truth table matches, so the two statements are true in exactly the same circumstances.",
            "The converse and inverse are equivalent to each other but not to the original, so a true conditional guarantees nothing about them. Assuming it does is the error behind most invalid geometric arguments."
          ],
          "hint": "Test each with a statement you know to be true but whose reverse is false — squares and rectangles work well."
        },
        {
          "id": "log-015",
          "topic": "Properties of Equality",
          "difficulty": "easy",
          "stem": "In a two-column proof, what reason justifies the step \"AB ≅ AB\"?",
          "choices": [
            "Symmetric Property",
            "Reflexive Property",
            "Transitive Property",
            "Segment Addition Postulate"
          ],
          "correctIndex": 1,
          "explanations": [
            "The Symmetric Property says that if a ≅ b then b ≅ a — it reverses a congruence you already have. Here nothing is being reversed; a single segment is being compared with itself.",
            "Correct. The Reflexive Property states that any quantity is congruent to itself. It looks trivial, but it is the reason a shared side counts as a pair of congruent sides when proving two triangles congruent by SSS or SAS.",
            "The Transitive Property chains two congruences: if a ≅ b and b ≅ c then a ≅ c. It needs two given statements to link, and here there is only one segment.",
            "The Segment Addition Postulate says that if B is between A and C then AB + BC = AC. It is about combining lengths along a segment, not about a segment's relationship to itself."
          ],
          "hint": "The step compares one object with itself. Which property is the one that permits that?"
        },
        {
          "id": "log-016",
          "topic": "CPCTC",
          "difficulty": "medium",
          "stem": "After proving △ABC ≅ △DEF, a proof concludes that ∠A ≅ ∠D. What reason justifies that step?",
          "choices": [
            "CPCTC",
            "ASA",
            "The Reflexive Property",
            "The definition of congruent angles"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. CPCTC — Corresponding Parts of Congruent Triangles are Congruent — is the reason that lets you harvest individual congruent sides and angles AFTER the triangles themselves have been proved congruent. The order matters: CPCTC is a consequence of congruence, never a way to establish it.",
            "ASA is a congruence criterion — one of the tools used to PROVE the triangles congruent in the first place. Using it here would be circular, since the triangle congruence is already established and is what the conclusion rests on.",
            "The Reflexive Property applies when an object is compared with itself. ∠A and ∠D belong to two different triangles, so nothing is being compared with itself.",
            "The definition of congruent angles says congruent angles have equal measure. That unpacks what ∠A ≅ ∠D means; it does not supply the reason the relationship holds here."
          ],
          "hint": "The triangles are already congruent. What licenses moving from whole-triangle congruence down to one pair of parts?"
        },
        {
          "id": "log-017",
          "topic": "Law of Syllogism",
          "difficulty": "hard",
          "stem": "Given \"If a figure is a square, then it is a rhombus\" and \"If a figure is a rhombus, then its diagonals are perpendicular,\" what does the Law of Syllogism let you conclude?",
          "choices": [
            "The figure is a square.",
            "If a figure is a square, then its diagonals are perpendicular.",
            "If a figure's diagonals are perpendicular, then it is a square.",
            "Nothing — the two statements are unrelated."
          ],
          "correctIndex": 1,
          "explanations": [
            "This would be the Law of Detachment, and it needs something the problem does not supply: a statement that some specific figure actually IS a square. Syllogism chains conditionals together; detachment fires a single conditional using a confirmed hypothesis.",
            "Correct. The Law of Syllogism chains p → q with q → r to give p → r. The conclusion of the first statement is exactly the hypothesis of the second, so the two link into a single conditional running from square to perpendicular diagonals.",
            "This reverses the chain, producing the converse of the correct conclusion. Perpendicular diagonals do not force a square — a kite has them too — so the reversed statement is actually false.",
            "They are precisely related: the conclusion of the first ('is a rhombus') matches the hypothesis of the second. That shared middle term is the hinge the Law of Syllogism turns on."
          ],
          "hint": "Look for a statement that appears as the conclusion of one conditional and the hypothesis of the other. That shared term is what gets eliminated."
        },
        {
          "id": "log-018",
          "topic": "Reasoning: Conditional Logic",
          "difficulty": "hard",
          "stem": "\"If a figure is a square, then it is a rectangle\" is true. A student concludes that \"if a figure is a rectangle, then it is a square\" must also be true. What is the error?",
          "choices": [
            "They formed the CONVERSE, which is not guaranteed by the original statement.",
            "They formed the contrapositive, which is never true.",
            "They negated only the hypothesis, forming the inverse, which is always true.",
            "There is no error; a conditional and its reverse always agree."
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Swapping hypothesis and conclusion gives the converse, and a true conditional says nothing about its converse. A long rectangle is a counterexample.",
            "The contrapositive swaps AND negates, and it is always equivalent to the original — it is the one form that is guaranteed true.",
            "The inverse negates both parts without swapping, and it is not guaranteed true either. This student swapped rather than negated.",
            "A conditional and its converse are independent. Only the contrapositive is logically equivalent to the original."
          ],
          "hint": "Which of the four related conditionals is the only one guaranteed to share the original's truth value?",
          "kind": "reasoning"
        },
        {
          "id": "log-019",
          "topic": "Reasoning: What's Missing",
          "difficulty": "medium",
          "stem": "A proof states: \"∠1 ≅ ∠2 because they are vertical angles.\" What must be true of the figure for this justification to hold?",
          "choices": [
            "The two angles must be supplementary.",
            "Two lines must actually intersect, with the angles opposite each other at that intersection.",
            "The angles must both be acute.",
            "The two angles must share a common side."
          ],
          "correctIndex": 1,
          "explanations": [
            "Supplementary angles sum to 180°. Vertical angles are congruent, which is a different relationship — and supplementary angles are usually not congruent.",
            "Correct. Vertical angles are defined by a crossing: two lines meeting at a point create two pairs of opposite angles. Without an actual intersection there are no vertical angles to appeal to.",
            "Vertical angles can be acute, right or obtuse. Their size is not part of the definition.",
            "Angles that share a side are ADJACENT. Vertical angles share only the vertex and lie opposite one another."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}