Conditionals
Identify hypothesis & conclusion in: "If it rains, then the ground is wet."
Answer: Hypothesis: "It rains"
Conclusion: "The ground is wet"
Define a biconditional statement in your own words.
A biconditional statement says that both the conditional and its converse are true, using "if and only if."
What makes a definition “good” in geometry?
A good definition is clear, precise, reversible (can be written as a biconditional), and free of vague or circular language.
What is the negation of “It is raining”?
"It is not raining."
Write the converse of: "If a figure is a square, then it has four sides."
"If a figure has four sides, then it is a square."
Write the biconditional for: "If a figure is a rectangle, then it has four right angles."
"A figure is a rectangle if and only if it has four right angles."
Identify the problem in this definition: “A triangle is a shape.”
It is too vague and incomplete; it does not describe what makes a triangle unique.
What is the difference between the inverse and contrapositive?
The inverse negates both hypothesis and conclusion; the contrapositive switches and negates both.
Write the inverse of: "If a number is even, then it is divisible by 2."
"If a number is not even, then it is not divisible by 2."
Is this biconditional true? "A figure is a rhombus if and only if it has four congruent sides." Explain.
Yes, it is true because a rhombus is defined as a quadrilateral with four congruent sides, and any quadrilateral with four congruent sides is a rhombus.
Rewrite this definition to make it better: “A square is a shape with four sides.”
"A square is a quadrilateral with four congruent sides and four right angles."
Given: “If a number is prime, then it has only two factors.” Write the contrapositive.
If a number does not have only two factors, then it is not prime."
Write the contrapositive of: "If a shape is a triangle, then it has 3 sides."
If a shape does not have 3 sides, then it is not a triangle."
Write a biconditional for: "If a polygon is regular, then all sides and angles are congruent."
"A polygon is regular if and only if all its sides and angles are congruent."
Explain why circular definitions are not good with an example.
Circular definitions use the term being defined in the definition itself, which doesn’t explain the term. Example: "A square is a square shape."
Explain why the contrapositive of a true conditional is always true.
Because the contrapositive is logically equivalent to the original conditional, they always share the same truth value.
True or False: The converse of a true conditional is always true. Explain.
False. The converse may or may not be true. For example, "If it is raining, then the ground is wet" is true, but the converse "If the ground is wet, then it is raining" is not always true (the ground could be wet for other reasons).
Explain why biconditional statements are important in geometry proofs.
Biconditionals allow precise definitions and equivalences, making proofs clearer and more rigorous by showing that two conditions imply each other.
Create a good definition for “parallelogram.”
A parallelogram is a quadrilateral with both pairs of opposite sides parallel."
Create your own conditional statement and write its converse, inverse, and contrapositive.
Example Answer:
Conditional: "If a figure is a pentagon, then it has five sides."
Converse: "If a figure has five sides, then it is a pentagon."
Inverse: "If a figure is not a pentagon, then it does not have five sides."
Contrapositive: "If a figure does not have five sides, then it is not a pentagon."