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Questions and Answers
What is a converse in terms of conditionals?
Which of the following statements is a counterexample to the converse of the conditional 'If I am a dinosaur, then I am extinct'?
How many counterexamples are needed to disprove a given statement?
What does the biconditional statement 'x + 3 = 5 if and only if x = 2' indicate?
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In the conditional 'If x + 3 = 5, then x = 2', what is the hypothesis?
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Which of the following statements is always true about angles?
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If points are collinear, then they lie in one plane.
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What does it mean if two angles are complementary?
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A right angle measures _____ degrees.
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Match the following terms with their definitions:
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What does the statement 'If ∠A is not obtuse, then ∠A is acute' imply?
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If angles are congruent, their measures can be different.
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The statement 'If ____ is between ____ and ____ then ____' expresses a condition about the order of points on a line.
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Study Notes
Conditionals
- Conditionals are if-then statements.
- The "if" part is called the hypothesis.
- The "then" part is called the conclusion.
Converses
- Converses are formed by interchanging the hypothesis and conclusion of a conditional.
- A conditional and its converse often express different ideas.
- Many true conditionals have false converses.
Counterexamples
- A counterexample disproves a given statement.
- Only one counterexample is necessary to disprove a statement.
Other Ways To Write a Conditional
- "If p, then q" can be written as:
- "p implies q"
- "p only if q"
- "q if p"
Biconditional
- A biconditional combines a true conditional and its true converse.
- It is written as "p if and only if q".
Angle and Line Relationships
- Right Angles: An angle is a right angle if and only if it measures 90 degrees.
- Straight Angles: An angle is a straight angle if and only if it measures 180 degrees.
- Collinear Points: Points are collinear if and only if they all lie on a single line.
- Coplanar Points: Points are coplanar if and only if they all lie within a single plane.
- Angle Congruence: Two angles are congruent if and only if they have the same measure.
Conditional Statements and Their Converse
- Conditional Statement: A conditional statement is a statement that can be written in the form "If p, then q," where p is the hypothesis and q is the conclusion.
- Converse: The converse of a conditional statement is formed by switching the hypothesis and conclusion. It takes the form "If q, then p."
- True and False Conditionals: A conditional statement is considered true if the conclusion is true whenever the hypothesis is true. If the hypothesis is true and the conclusion is false, then the statement is false.
Examples of Conditional Statements and Their Truth Values
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Statement: If a figure is a square, then it is a rectangle.
- Converse: If a figure is a rectangle, then it is a square.
- Truth Value: The original statement is true, the converse is false.
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Statement: If two angles are congruent, then their measures are equal.
- Converse: If the measures of two angles are equal, then the angles are congruent.
- Truth Value: Both original statement and the converse are true.
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Statement: If a point is the midpoint of a segment, then it divides the segment into two congruent segments.
- Converse: If a point divides a segment into two congruent segments, then it is the midpoint of the segment.
- Truth Value: Both original statement and its converse are true.
-
Statement: If an angle is obtuse, then it is not a right angle.
- Converse: If an angle is not obtuse, then it is a right angle.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If lines are parallel, then they do not intersect.
- Converse: If lines do not intersect, then they are parallel.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If a number is even, then it is divisible by 2.
- Converse: If a number is divisible by 2, then it is even.
- Truth Value: Both original statement and the converse are true.
-
Statement: If a number is divisible by 4, then it is divisible by 2.
- Converse: If a number is divisible by 2, then it is divisible by 4.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If today is Friday, then tomorrow is Saturday.
- Converse: If tomorrow is Saturday, then today is Friday.
- Truth Value: Both original statement and the converse are true.
-
Statement: If is a triangle, then its angles add up to 180 degrees.
- Converse: If the angles of a figure add up to 180 degrees, then it is a triangle.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If Pam lives in Chicago, then she lives in Illinois.
- Converse: If Pam lives in Illinois, then she lives in Chicago.
- Truth Value: The original statement is true, the converse is false.
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Statement: If and , then ∠≅
- Converse: If ∠≅ , then and .
- Truth Value: The original statement and its converse are true.
-
Statement: " " if and only if and "
- Converse: " and " if and only if " "
- Truth Value: Both the original statement and its converse are true.
-
Statement: " " only if "
- Converse: " " only if "
- Truth Value: The original statement is true, the converse is false.
-
Statement: " " only if "
- Converse: " " only if "
- Truth Value: The original statement is false, the converse is true.
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Statement: If a triangle is an equilateral triangle then it is an isosceles triangle.
- Converse: If a triangle is an isosceles triangle then it is an equilateral triangle.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If a dog is a Labrador Retriever, then it is a dog.
- Converse: If a dog is a dog, then it is a Labrador Retriever.
- Truth Value: The original statement is true, the converse is false.
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Statement: If a number is odd, then it is not divisible by 2.
- Converse: If a number is not divisible by 2, then it is odd.
- Truth Value: The original statement is true, the converse is true.
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Statement: If a quadrilateral is a square, then it is a rhombus.
- Converse: If a quadrilateral is a rhombus, then it is a square.
- Truth Value: The original statement is true, the converse is false.
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Statement: If a triangle is a right triangle, then its angles add up to 180 degrees.
- Converse: If the angles of a triangle add up to 180 degrees, then it is a right triangle.
- Truth Value: The original statement is true, the converse is false.
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Statement: If , , are collinear, then they all lie on one line
- Converse: If are collinear, then , , are collinear
- Truth Value: Both the original statement and its converse are true.
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Statement: If ! " , then " is the midpoint of !
- Converse: If " is the midpoint of !, then ! "
- Truth Value: The original statement is true, the converse is false.
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Statement: If a quadrilateral is a rectangle, then it has four right angles.
- Converse: If a quadrilateral has four right angles, then it is a rectangle.
- Truth Value: The original statement is true, the converse is false.
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Statement: If is a triangle, then the sum of its interior angles is .
- Converse: If the sum of the interior angles of a figure is , then the figure is a triangle.
- Truth Value: The original statement is true, the converse is false.
-
Statement: If a figure is a rectangle, then it has two pairs of parallel sides.
- Converse: If a figure has two pairs of parallel sides, then it is a rectangle.
- Truth Value: The original statement is true, the converse is false.
Biconditionals
- Biconditional Statement: A biconditional statement combines a conditional statement and its converse. It is written in the form "p if and only if q." It is true if both the original statement and its converse are true.
-
Example: "A triangle is an equilateral triangle if and only if it is an isosceles triangle with all sides equal."
- Truth Value: This biconditional statement is false.
Important Points to Remember
- The truth value of a conditional statement is not the same as the truth value of its converse.
- A true statement can have a false converse, and vice versa.
- A biconditional statement is only true if both the original statement and its converse are true.
- "If and only if" statements are biconditionals.
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Description
Test your understanding of conditionals, converses, and biconditionals with this quiz. Explore the nuances of truth values in logical statements and the importance of counterexamples in disproving arguments.
