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

Grade 10 Math Worksheet - Building Confidence

Circle the best answer.

1. 1.

   Ari claims: every quadrilateral with four equal sides is a square. Which shape is a counterexample?

   • A rectangle that is 3 by 5
   • A square with side 4 and four right angles, drawn on the grid
   • A rhombus with side 4 and angles of 60° and 120°
   • An equilateral triangle with side 4
2. 2.

   A counterexample has two angles adding to 180° where one is not acute or one is not obtuse. Bo claims: if two angles add to 180°, one is acute and one is obtuse. Which pair is a counterexample?

   • 90° and 90°
   • 40° and 140°
   • Two acute angles that add to 90°
   • 30° and 60°
3. 3.

   a counterexample makes n² greater than 25 while n is not greater than 5. Esme claims: if n² > 25, then n > 5. Which value of n is a counterexample?

   • n = 4
   • n = 7
   • n = -6
   • n = -5
4. 4.

   Two diagonals bisect each other when each cuts the other at its midpoint. Jun claims: the diagonals of every quadrilateral bisect each other. Which shape is a counterexample?

   • Any quadrilateral whose diagonals are 5 and 7 long
   • A kite whose diagonals do not bisect each other
   • A triangle with three sides
   • A rectangle that is 4 by 6, with equal diagonals
5. 5.

   A negation is true exactly when the original statement is false. Kira says: all the pencils in the box are sharp. Which statement is the negation?

   • No pencil in the box is sharp, not even one of them
   • Some pencils in the box are sharp
   • The box has pencils in it
   • At least one pencil in the box is not sharp
6. 6.

   The rule is x > 5. The negation is true for exactly the numbers that do not satisfy the rule. A rule says x > 5. Luz writes the negation. Which statement is it?

   • x ≤ 5
   • x > -5
   • x = 5
   • x < 5
7. 7.

   The statement joins two facts with 'and'. Milo says: the triangle is right and isosceles. Which statement is its negation?

   • The triangle is not right
   • The triangle is acute and scalene
   • The triangle is not right or it is not isosceles
   • The triangle is not right and it is not isosceles
8. 8.

   p is true. q is false. p: a triangle has 3 sides (true). q: a square has 5 sides (false). Noah checks four statements. Which one is true?

   • p and q
   • p or q
   • not (p or q)
   • not p
9. 9.

   In Ren's class of 40, 22 play soccer, 15 play chess and 6 play both. How many play exactly one of the two?

   • 25
   • 31
   • 16
   • 37
10. 10.

    In Zee's class of 40, how many students play soccer but not chess?

    • 17
    • 27
    • 10
    • 7