The inverse of a conditional statement is, "If ABC is a triangle, then ∠A+∠B+∠C=180^∘ ". Then th

step 1 So here we are given the inverse of a conditional statement. That is, if A, B, C is a triangle,

The inverse of a conditional statement is, "If ABC is a...

The inverse of a conditional statement is, "If $\mathrm{ABC}$ is a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$ ". Then the contrapositive of the conditional statement is (1) "If $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$, then $\mathrm{ABC}$ is a triangle. (2) "If $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C} \neq 0$, then $\mathrm{ABC}$ is not a triangle. (3) "If $\mathrm{ABC}$ is not a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C} \neq 180^{\circ}$. (4) "If $\mathrm{ABC}$ is a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$.

The inverse of a conditional statement is, "If $\mathrm{ABC}$ is a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$ ". Then the contrapositive of the conditional statement is
(1) "If $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$, then $\mathrm{ABC}$ is a triangle.
(2) "If $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C} \neq 0$, then $\mathrm{ABC}$ is not a triangle.
(3) "If $\mathrm{ABC}$ is not a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C} \neq 180^{\circ}$.
(4) "If $\mathrm{ABC}$ is a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$.

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Key Concepts

Conditional Statement

A conditional statement is a logical statement of the form 'if P then Q', where P is the hypothesis (or antecedent) and Q is the conclusion (or consequent). This type of statement asserts that whenever P is true, Q must also be true, and it is widely used in proofs and logical reasoning.

Contrapositive

The contrapositive of a conditional statement 'if P then Q' is the statement 'if not Q then not P'. This reformulation is logically equivalent to the original conditional, meaning that both statements share the same truth value in every possible scenario. Recognizing and using the contrapositive can often simplify proofs and understanding of logical implications.

Logical Equivalence

Logical equivalence refers to the relationship between two statements that are true in exactly the same situations. In the context of conditional statements, the original statement and its contrapositive are considered logically equivalent, which is a key concept in logical reasoning and mathematical proofs.

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