Question

"If two triangles are congruent, then their areas are equal." is the given statement, then the contrapositive of the inverse of the given statement is

• A. If two triangles are not congruent, then their areas are equal.
• B. If two triangles are not congruent, then their areas are not equal.
• C. If areas of two triangles are equal, then they are congruent.
• D. If areas of two triangles are not equal, then they are congruent.

Hint:

Here is an amazing shortcut in mathematical logic: taking the contrapositive of any statement flips its order but preserves its logical truth value. The contrapositive of an inverse ($\sim p \rightarrow \sim q$) will always simplify right back to the original converse statement ($q \rightarrow p$)!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a conditional mathematical implication statement. We need to find the symbolic logical form of its inverse, determine the contrapositive of that inverse statement, and translate it back into natural language.

Step 2: Key Formula or Approach:
Let's assign simple statement variables to the component propositions:
Let $p$: Two triangles are congruent.
Let $q$: Their areas are equal.
The given conditional statement is written symbolically as: $p \rightarrow q$.
The logical rules governing these transformations are:

Inverse of $(p \rightarrow q)$ is defined as $\sim p \rightarrow \sim q$.

Contrapositive of a conditional statement $(A \rightarrow B)$ is defined as $\sim B \rightarrow \sim A$.

Step 3: Detailed Explanation:
Let's apply these steps sequentially:
1. Write down the inverse of our base conditional statement:
$$\text{Inverse} = \sim p \rightarrow \sim q$$ 2. Now, construct the contrapositive of this newly formed inverse statement by swapping the hypothesis and conclusion components and negating them both:
$$\text{Contrapositive of Inverse} = \sim(\sim q) \rightarrow \sim(\sim p)$$ 3. According to the law of double negation, $\sim(\sim q) \equiv q$ and $\sim(\sim p) \equiv p$. Therefore, the statement simplifies perfectly to:
$$q \rightarrow p$$ Translating the conditional $q \rightarrow p$ back into an English sentence gives:
"If areas of two triangles are equal, then they are congruent."
This matches option (C).

Step 4: Final Answer:
The contrapositive of the inverse statement is "If areas of two triangles are equal, then they are congruent.", which corresponds to option (C).