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

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   A counterexample has perpendicular diagonals but is not a rhombus. Ari claims: if a quadrilateral's diagonals are perpendicular, it is a rhombus. Which figure is a counterexample?

   • A 3 by 4 rectangle, whose diagonals are not perpendicular
   • A kite with perpendicular diagonals and sides 3, 3, 5, 5
   • A rhombus with diagonals 6 and 8
   • A square with side 4 and perpendicular diagonals
2. 2.

   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)
   • (0, 0), (4, 0), (2, 3)
   • (1, 1), (2, 2), (3, 3)
3. 3.

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

4. Read this first

   Use this

   "P if and only if Q" means both "if P then Q" and "if Q then P". P: a quadrilateral is a rectangle. Q: it has four right angles.

5. 4.

   A definition says: a quadrilateral is a rectangle if and only if it has four right angles. Kira unpacks it. Which statement says the same thing?

   • If a quadrilateral has four right angles, then it is a square, and a square is a special rectangle
   • If a quadrilateral is a rectangle, it has four right angles; if it is not a rectangle, it does not
   • If a quadrilateral is a rectangle, then it has four right angles; the other direction is not claimed
   • A rectangle has four right angles, and a quadrilateral with four right angles is a rectangle

6. The proof's step

   "diagonals not congruent, so not a rectangle". The contrapositive of "if P then Q" is "if not Q then not P".

7. 5.

   A proof says: the diagonals of ABCD are not congruent, so ABCD is not a rectangle. Milo asks which theorem it uses as its contrapositive. Which one?

   • If a quadrilateral is not a rectangle, then its diagonals are not congruent
   • If a quadrilateral is a rectangle, then its diagonals are congruent
   • If a quadrilateral is a rectangle, then it is a parallelogram
   • If a quadrilateral's diagonals are congruent, then it is a rectangle
8. 6.

   Pia studies the theorem: if a quadrilateral is a parallelogram, then its opposite sides are congruent. Which statement about its converse is correct?

   • The converse is also true, so together they describe parallelograms
   • The inverse is the one that is true, and the converse is not true here
   • The converse is false, because a kite has two pairs of congruent sides and is not a parallelogram
   • The converse is false, because the converse of a true theorem is a false statement
9. 7.

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

   • Every rectangle is a rhombus, because both circles sit in the same parallelogram circle
   • 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 square is both a rectangle and a rhombus, so squares sit in the overlap
10. 8.

    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
    • True: every quadrilateral with perpendicular diagonals is a rhombus
    • 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
11. 9.

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

    • An isosceles trapezoid: congruent diagonals and no right angles
    • A rhombus that is not a square, because its diagonals differ in length
    • A rectangle, because it has congruent diagonals
    • A square, because its diagonals are perpendicular and congruent
12. 10.

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

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