Questions and Answers
Which are true regarding the statement 'a quadrilateral having exactly one pair of parallel opposite sides is a trapezoid'? (Select all that apply)
What is the converse of the statement 'if a line is vertical then it has undefined slope'?
If a line has an undefined slope then it is a vertical line
What is the inverse of the statement 'if a line is vertical then it has undefined slope'?
If a line is not vertical then it does not have an undefined slope
What is the contrapositive of the statement 'if a line is horizontal then it has slope equal to zero'?
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What is the contrapositive of the statement 'if there is a lightning storm then we won't go swimming'?
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What is a true conditional statement based on the diagram?
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Match the conditional statement to its symbolic form:
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Determine the truth value of the statement: 'If 3 is an even number then 5 + 3 = 8'.
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What is equivalent to ~Q → ~P if P: N is a multiple of two and Q: N is an even number?
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Identify the truth table for Q → P where Q represents a statement.
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Which statement represents a true conclusion based on the Venn diagram?
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The hypothesis of the statement 'if the last digit of N is zero then N is divisible by five' is: ______
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What is the conclusion of the statement: 'if the last digit of N is zero then N is divisible by five'?
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What is the converse of 'if Jack's pet is a pig then Jack's pet cannot fly'?
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What is the logically equivalent statement to Q → P where P is 'student achieves 90 percent on a geometry final' and Q is 'student will receive a passing grade in geometry class'?
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What is the contrapositive of 'if 3x + 7 = -5 then 4x + 12 = -4'?
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What is the logically equivalent statement for the conditional statement P → ~Q?
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Which pairs of statements are logically equivalent? (Select 2 options)
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What is the inverse of P → Q?
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What is the contrapositive of the original conditional statement represented by P → Q?
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What is the contrapositive of the statement: 'if two variables are directly proportional then their graph is a linear function'?
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What represents the inverse in the original conditional statement P → Q?
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What is the alternative name for the inverse of a conditional statement?
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What is the converse of 'if x is even then x + 1 is odd'?
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Study Notes
Quadrilaterals and Trapezoids
- A trapezoid is a quadrilateral with exactly one pair of parallel opposite sides.
- Conditional statement format: "If a quadrilateral has exactly one pair of parallel opposite sides, then it is a trapezoid."
- In this conditional statement:
- Hypothesis: "A quadrilateral having exactly one pair of parallel opposite sides."
- Conclusion: "It is a trapezoid."
Converse and Inverse Statements
- The converse of a statement flips the hypothesis and conclusion.
- For "If a line is vertical, then it has undefined slope," the converse is: "If a line has an undefined slope, then it is vertical."
- The inverse negates both the hypothesis and conclusion.
- For the same statement, the inverse is: "If a line is not vertical, then it does not have an undefined slope."
Contrapositive
- The contrapositive negates both parts and swaps them.
- For "If a line is horizontal, then it has a slope equal to zero," the contrapositive is: "If a line does not have zero slope, then it is not horizontal."
Conditional Statement Example
- The conditional "If there is a lightning storm, then we won't go swimming" has the contrapositive: "If we go swimming, then there will not be a lightning storm."
True Conditional Statement
- Based on a given diagram, a true statement could be: "If a student is an honor student, then he/she may also be a band member."
Symbolic Representation of Conditions
- Statements can be represented symbolically:
- ~p → ~q: "If today is not October 6, then it is not Nina's birthday."
- q → p: "If today is Nina’s birthday, then it is October 6."
- p → q: "If today is October 6, then it is Nina's birthday."
- ~q → ~p: "If it is not Nina's birthday, then today is not October 6."
Truth Value of Conditional Statements
- A false statement (e.g., "3 is an even number") implies a true statement, making the overall conditional false.
Equivalent Statements
- The logic behind statements: ~Q → ~P is the inverse of P → Q, indicating that if N is not even, then it is not a multiple of two, which is equivalent to P → Q.
Converse and Inverse Evaluation
- The inverse of "If you are human, then you were born on Earth" is true: "If you are not human, then you were not born on Earth."
- The converse "If you were born on Earth, then you are human" is false because nonhuman creatures also exist on Earth.
Truth Tables
- Based on the conditional expression Q → P, a truth table categorizes outcomes: TTFF, TFTF, TTFT.
Venn Diagram Conclusions
- According to a Venn diagram, some numbers that are multiples of three are also even.
Statements and Their Relations
- The relationship P: "Student achieves 90% on a geometry final" and Q: "Student will receive a passing grade in geometry class" shows that Q → P implies students not passing did not score 90%.
Logical Equivalence
- In terms of logical equivalency, P → Q is equivalent to ~Q → ~P; both convey similar information about the statements' relationships.
Overall Connections
- Understanding statements, converses, inverses, and contrapositives are critical in logical reasoning and geometry.
- The equivalency of statements can often assist in deducing the truth of conditions based on given scenarios or hypotheses.
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Description
Test your understanding of trapezoids with this flashcard quiz. Focus on the conditional statements related to quadrilaterals and their properties. Check your knowledge on what defines a trapezoid and its characteristics.
