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

Answer

To find the negation of the statement "ABC is an equilateral triangle if and only if it is equiangular," we can follow these steps:

### Step 1: Identify the components of the biconditional statement
The statement can be represented as:
- Let \( p \): "ABC is an equilateral triangle."
- Let \( q \): "ABC is equiangular."

The original statement can be written as:
\[ p \iff q \]

### Step 2: Understand the negation of a biconditional statement
The negation of a biconditional statement \( p \iff q \) is given by:
\[ \neg(p \iff q) \]
This can be expressed as:
\[ (p \land \neg q) \lor (\neg p \land q) \]

### Step 3: Write the negation in terms of \( p \) and \( q \)
Using the definitions of \( p \) and \( q \):
- \( \neg p \): "ABC is not an equilateral triangle."
- \( \neg q \): "ABC is not equiangular."

Thus, the negation can be written as:
\[ (p \land \neg q) \lor (\neg p \land q) \]
This translates to:
- \( (ABC \text{ is an equilateral triangle} \land ABC \text{ is not equiangular}) \lor (ABC \text{ is not an equilateral triangle} \land ABC \text{ is equiangular}) \)

### Step 4: Write the final negation statement
The final negation statement is:
"ABC is an equilateral triangle and ABC is not equiangular, or ABC is not an equilateral triangle and ABC is equiangular."

### Summary of the Negation
The negation of the statement "ABC is an equilateral triangle if and only if it is equiangular" is:
"ABC is an equilateral triangle and ABC is not equiangular, or ABC is not an equilateral triangle and ABC is equiangular."

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