Counterexamples, "not", "and", "or", and Venn diagrams
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
Proof by contradiction starts by assuming the opposite of what is to be shown, then reaches a contradiction. Esme proves 'a triangle cannot have two right angles' by contradiction. Which assumption should the proof start from?
- Assume the angle sum is 180°, then show a triangle has two right angles
- Assume a triangle has two right angles, then show the sum exceeds 180°
- Assume a triangle has three right angles, then look for a contradiction
- Assume a triangle has no right angle, then show the angle sum is not 180°, a contradiction
A counterexample has perimeter 24 and area greater than 35. Ari claims: if a rectangle has perimeter 24, its area is at most 35. Which rectangle is a counterexample?
- A 6 by 7 rectangle, with area 42
- A 4 by 8 rectangle, with area 32
- A 5 by 7 rectangle, with area 35
- A 6 by 6 square, with area 36
Of Bo's 60 classmates, 35 take geometry, 28 take art and 12 take neither. How many take both?
- 3
- 48
- 15
- 20
Read this first
Use this
Art 20, Band 18, Chess 15. Art and Band 7, Art and Chess 6, Band and Chess 5, each including the students who like all three. All three: 3.
Jun surveys students about Art, Band and Chess. Using the counts on the card, how many like at least one?
- 38
- 53
- 32
- 35
S is true. Two statements are equivalent when they are true or false together. S: if a quadrilateral has four equal sides, then its diagonals are perpendicular. Milo wants a true statement equivalent to S. Which one?
- If a quadrilateral does not have four equal sides, then its diagonals are not perpendicular
- If a quadrilateral's diagonals are perpendicular, then it has four equal sides
- If a quadrilateral's diagonals are not perpendicular, then its four sides are not all equal
- A quadrilateral has four equal sides and its diagonals are not perpendicular
Use this
The contrapositive of 'if P then Q' is 'if not Q then not P'. P: n² is even. Q: n is even.
Noah proves 'if n² is even, then n is even' by contrapositive. Which should the proof start from?
- Assume n² is odd, then show n is odd
- Assume n² is even, then show n is even
- Assume n is even, then show n² is even
- Assume n is odd, then show n² is odd
Total 80. Soccer 45, basketball 40, both 12, neither 5. Zee reads a survey of 80 students: 45 like soccer, 40 like basketball, 12 like both and 5 like neither. Does the data fit together?
- Yes: each count is less than 80
- Yes: 45 + 40 + 5 − 12 = 78, which is close to 80
- No: 45 + 40 − 12 = 73 like at least one, but 80 − 5 = 75 should
- No: 45 + 40 = 85 is more than the 80 students, so the data is impossible
For every triangle, at least one side is shorter than 10. Ren says: for every triangle there is a side shorter than 10. Which statement is its negation?
- For every triangle there is a side longer than 10
- For every triangle, all sides are at least 10
- There is a triangle whose sides are all at least 10
- There is a triangle with a side longer than 10
A counterexample has equal diagonals and is not a rectangle. Luz claims: if a quadrilateral's diagonals have equal length, it is a rectangle. Which quadrilateral is a counterexample?
- Corners (0, 0), (6, 0), (5, 3), (1, 3): equal diagonals √34, no right angles
- Corners (0, 0), (4, 0), (4, 4), (0, 4): equal diagonals √32, a square
- Corners (0, 0), (6, 0), (8, 3), (2, 3): the diagonals are √73 and 5
- Corners (0, 0), (6, 0), (6, 3), (0, 3): equal diagonals √45, a rectangle
A new club survey has Chess only 0, Art+Chess only 3, Band+Chess only 2, all three 1. Pia claims everyone in Chess is also in Art or Band. Is it true?
- True: the Chess-only region is empty, so every Chess member is in Art or Band
- False: the 9 students in no club break the claim about Chess members
- False: Chess has 6 members, and only the 3 in Art and Chess count as covered
- True: Chess has fewer members than Art, so they must sit inside Art's circle