Counterexamples, "not", "and", "or", and Venn diagrams

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   Three points form a triangle unless they lie on one line. Bo claims: any three points with different x-coordinates are corners of a triangle. Which set is a counterexample?

   • (1, 2), (3, 6), (5, 9)
   • (0, 0), (0, 5), (3, 5)
   • (1, 1), (2, 2), (3, 3)
   • (0, 0), (4, 0), (2, 3)
2. 2.

   The contrapositive of 'if P then Q' is 'if not Q then not P'. Statement: if a figure is a square, then it has four right angles. Esme writes the contrapositive. Which one is it?

   • If a figure is not a square, then it has four right angles
   • If a figure has four right angles, then it is a square
   • If a figure is not a square, then it does not have four right angles
   • If a figure does not have four right angles, then it is not a square
3. 3.

   Both the statement and its converse are true. Every rhombus has perpendicular diagonals. A kite can have perpendicular diagonals. Esme proposes a definition: a quadrilateral is a rhombus if and only if its diagonals are perpendicular. Is it a good definition?

   • No: a rhombus does not have perpendicular diagonals, so the statement itself is false and fails
   • Yes: a definition needs just the direction that is true for rhombi, and that direction holds
   • Yes: a rhombus has perpendicular diagonals, so the converse follows as well, and the definition works
   • No: the converse fails, as a kite has perpendicular diagonals and is no rhombus

4. Read this first

   Use this

   The claim says every quadrilateral with congruent diagonals is a rectangle. The negation is true exactly when the claim is false.

5. 4.

   A student claims: every quadrilateral with congruent diagonals is a rectangle. Noah writes the negation. Which statement is it?

   • Some rectangle has diagonals that are not congruent to each other
   • No quadrilateral with congruent diagonals is a rectangle outright
   • Each rectangle has congruent diagonals, which is a known theorem
   • Some quadrilateral has congruent diagonals but is not a rectangle
6. 5.

   Ren draws a Venn diagram of quadrilaterals: rectangles and rhombi are circles inside parallelograms. Which claim does the diagram support?

   • Every square is both a rectangle and a rhombus, so squares sit in the overlap
   • Rectangles and rhombi share no shapes, because they are different circles
   • Some rhombus is not a parallelogram, since rhombi reach outside the large circle
   • Every rectangle is a rhombus, because both circles sit in the same parallelogram circle
7. 6.

   Tala claims: every parallelogram with perpendicular diagonals is a rhombus. Using the family Venn diagram, what is the verdict?

   • False: a square has perpendicular diagonals and is not a rhombus, so the claim fails
   • False: a kite has perpendicular diagonals and is not a rhombus, so the claim fails
   • True: the four right triangles are congruent, so every side is congruent
   • True: every quadrilateral with perpendicular diagonals is a rhombus
8. 7.

   Zee claims: a quadrilateral with congruent diagonals is a rectangle. Which figure is a counterexample, and why?

   • A rectangle, because it has congruent diagonals
   • A rhombus that is not a square, because its diagonals differ in length
   • An isosceles trapezoid: congruent diagonals and no right angles
   • A square, because its diagonals are perpendicular and congruent
9. 8.

   Jun claims: no rectangle is a rhombus. Using the family Venn diagram, what is the verdict?

   • False: a square is both, so it sits in the overlap
   • False: a parallelogram with unequal sides is both a rectangle and a rhombus
   • True: the two circles have different names
   • True: a rectangle has unequal sides, so it cannot also be a rhombus
10. 9.

    Kira claims: everyone in Art is also in Band or Chess. Using the 50-student counts, what does the diagram show?

    • False: the 8 students who are in Art only break the claim
    • True: 8 of Art's 16 students are in another club, which is most of them
    • True: Band and Chess together have more members than Art has
    • False: the 21 students in no club break the claim
11. 10.

    Is "if not P then not Q". P: a triangle is equilateral. Q: it is equiangular. Theorem: if a triangle is equilateral, then it is equiangular. Pia writes the inverse. Which statement is it?

    • If a triangle is not equiangular, then it is not equilateral
    • A triangle is equilateral and it is not equiangular
    • If a triangle is not equilateral, then it is not equiangular
    • If a triangle is equiangular, then it is equilateral