Write the negatin of each of the statement :
ABC is an equilateral triangle if and only if it is equiangular.

Answer

To negate the statement "ABC is an equilateral triangle if and only if it is equiangular," we will break it down into its components and apply logical reasoning.

### Step-by-Step Solution:

1. **Identify the Components of the Statement**:
- Let \( P \) be the statement "ABC is an equilateral triangle."
- Let \( Q \) be the statement "ABC is an equiangular triangle."

The original statement can be expressed as:
\[ P \iff Q \]
This means "P if and only if Q."

2. **Understand the Negation of the Biconditional**:
The negation of a biconditional statement \( P \iff Q \) is given by:
\[ \neg(P \iff Q) \equiv P \land \neg Q \lor \neg P \land Q \]
This means that either \( P \) is true and \( Q \) is false, or \( P \) is false and \( Q \) is true.

3. **Write the Negation**:
From the above understanding, we can write the negation of the original statement:
\[ \text{"ABC is an equilateral triangle and not equiangular, or ABC is not equilateral and is equiangular."} \]
In simpler terms:
\[ (ABC \text{ is equilateral and not equiangular}) \lor (ABC \text{ is not equilateral and is equiangular}) \]

### Final Negated Statement:
Thus, the negation of the statement "ABC is an equilateral triangle if and only if it is equiangular" is:
- "ABC is an equilateral triangle and not equiangular, or ABC is not equilateral and is equiangular."