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 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
   • 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
2. 2.

   If a quadrilateral has four equal sides, then it is a square. 'If a quadrilateral is a square, then it has four equal sides' is true. Jun tests its converse. What is correct?

   • The converse is true, because the converse of a true statement is also true
   • The converse is true: a square with side 5 has four equal sides and is a square
   • The converse is true, because squares have four equal sides, which is what the original says
   • The converse is false: a rhombus with 60° and 120° angles has four equal sides
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?

   • 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
   • Yes: a definition needs just the direction that is true for rhombi, and that direction holds
   • No: a rhombus does not have perpendicular diagonals, so the statement itself is false and fails

4. 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".

5. 4.

   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 a rectangle, then it is a parallelogram
   • 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

6. Use this

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

7. 5.

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

   • Some quadrilateral has congruent diagonals but is not a rectangle
   • 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
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 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
   • The converse is also true, so together they describe parallelograms
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
   • 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
10. 8.

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

    • False: a kite has perpendicular diagonals and is not a rhombus, so the claim fails
    • False: a square 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
11. 9.

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

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