Question

select the correct answer.
statement 1: a figure is a triangle if and only if it has three sides.
statement 2: a figure has three sides if and only if it is a triangle.
how is statement 2 related to statement 1?
a. statement 2 is the contrapositive of statement 1.
b. statement 2 is the inverse of statement 1.
c. statement 2 is the converse of statement 1.
d. statement 2 is the biconditional of statement 1.

Explanation:

First, recall the definitions:

• Biconditional: A statement "P if and only if Q" (P ↔ Q) is equivalent to (P → Q) ∧ (Q → P), meaning both the original and its converse are true.
• Statement 1: "A figure is a triangle if and only if it has three sides" (P ↔ Q, where P = "is a triangle", Q = "has three sides").
• Statement 2: "A figure has three sides if and only if it is a triangle" (Q ↔ P, which is logically equivalent to P ↔ Q, as biconditionals are symmetric).

Now analyze the options:

• Option A (Contrapositive): Contrapositive of P→Q is ¬Q→¬P. Statement 2 is not a contrapositive.
• Option B (Inverse): Inverse of P→Q is ¬P→¬Q. Statement 2 is not an inverse.
• Option C (Converse): Converse of P→Q is Q→P. But Statement 1 is a biconditional (P↔Q), and Statement 2 is also a biconditional (Q↔P), not just a converse.
• Option D (Biconditional): Since P↔Q is equivalent to Q↔P, Statement 2 is the biconditional form of Statement 1 (just swapping P and Q, which does not change the biconditional’s truth value).

Answer:

D. Statement 2 is the biconditional of statement 1.