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

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   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 does not have four right angles, then it is not a square
   • 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 is not a square, then it has four right angles
2. 2.

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

3. Read this first

   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".

4. 3.

   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's diagonals are congruent, then it is a rectangle
   • If a quadrilateral is a rectangle, then it is a parallelogram

5. Use this

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

6. 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
   • Some quadrilateral has congruent diagonals but is not a rectangle
   • Each rectangle has congruent diagonals, which is a known theorem
   • No quadrilateral with congruent diagonals is a rectangle outright
7. 5.

   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 converse is false, because the converse of a true theorem is a false statement
   • The converse is false, because a kite has two pairs of congruent sides and is not a parallelogram
   • The inverse is the one that is true, and the converse is not true here
8. 6.

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

   • Rectangles and rhombi share no shapes, because they are different circles
   • Every rectangle is a rhombus, because both circles sit in the same parallelogram circle
   • Every square is both a rectangle and a rhombus, so squares sit in the overlap
   • Some rhombus is not a parallelogram, since rhombi reach outside the large circle
9. 7.

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

   • A rhombus that is not a square, because its diagonals differ in length
   • A rectangle, because it has congruent diagonals
   • An isosceles trapezoid: congruent diagonals and no right angles
   • A square, because its diagonals are perpendicular and congruent
10. 8.

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

    • True: a rectangle has unequal sides, so it cannot also be a rhombus
    • False: a parallelogram with unequal sides is both a rectangle and a rhombus
    • True: the two circles have different names
    • False: a square is both, so it sits in the overlap
11. 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
    • False: the 21 students in no club break the claim
    • True: Band and Chess together have more members than Art has
    • True: 8 of Art's 16 students are in another club, which is most of them
12. 10.

    The negation of 'if P then Q' is 'P and not Q'. Statement: if a polygon is a hexagon, then it has 6 sides. Luz writes the negation of this statement. Which one is it?

    • If a polygon has 6 sides, then it is a hexagon
    • If a polygon is a hexagon, then it does not have 6 sides
    • A polygon is a hexagon and does not have 6 sides
    • If a polygon is not a hexagon, then it does not have 6 sides