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 rhombus with diagonals 6 and 8
   • A 3 by 4 rectangle, whose diagonals are not perpendicular
   • A square with side 4 and perpendicular diagonals
   • A kite with perpendicular diagonals and sides 3, 3, 5, 5
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), (4, 0), (2, 3)
   • (0, 0), (0, 5), (3, 5)
   • (1, 1), (2, 2), (3, 3)
3. 3.

   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 false: a rhombus with 60° and 120° angles has four equal sides
   • The converse is true, because squares have four equal sides, which is what the original says
   • The converse is true: a square with side 5 has four equal sides and is a square
   • The converse is true, because the converse of a true statement is also true
4. 4.

   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 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
   • No: the converse fails, as a kite has perpendicular diagonals and is no rhombus
5. 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 false, because the converse of a true theorem is a false statement
   • 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
   • The converse is false, because a kite has two pairs of congruent sides and is not a parallelogram
6. 6.

   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
   • Every rectangle is a rhombus, because both circles sit in the same parallelogram circle
   • Some rhombus is not a parallelogram, since rhombi reach outside the large circle
7. 7.

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

   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
9. 9.

   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 is a hexagon, then it does not have 6 sides
   • If a polygon has 6 sides, then it is a hexagon
   • If a polygon is not a hexagon, then it does not have 6 sides
   • A polygon is a hexagon and does not have 6 sides
10. 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
    • If a triangle is equiangular, then it is equilateral
    • A triangle is equilateral and it is not equiangular
    • If a triangle is not equilateral, then it is not equiangular