Write the negation of the statement "An angle is a right angle if and only if it is of measure `90^(@)`"

Answer

To write the negation of the statement "An angle is a right angle if and only if it is of measure 90°," we need to understand the structure of the statement and how to negate it.

### Step-by-Step Solution:

1. **Identify the Original Statement**:
The statement "An angle is a right angle if and only if it is of measure 90°" can be expressed in logical terms as:
\[ P \iff Q \]
where \( P \) is "An angle is a right angle" and \( Q \) is "It is of measure 90°."

2. **Understand the Meaning of "If and Only If"**:
The phrase "if and only if" indicates that both statements are equivalent. This means:
- If \( P \) is true, then \( Q \) is true.
- If \( Q \) is true, then \( P \) is true.
Therefore, the statement can be split into two implications:
\[ P \implies Q \quad \text{and} \quad Q \implies P \]

3. **Negate the Statement**:
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.

4. **Translate Back to the Context**:
- \( P \) is "An angle is a right angle."
- \( Q \) is "It is of measure 90°."

Therefore, the negation can be expressed as:
- "An angle is a right angle and it is not of measure 90°" (i.e., \( P \land \neg Q \))
- "An angle is not a right angle and it is of measure 90°" (i.e., \( \neg P \land Q \))

5. **Final Negation Statement**:
Thus, the complete negation of the original statement can be summarized as:
- "An angle is a right angle and it is not of measure 90° or an angle is of measure 90° and it is not a right angle."

### Final Answer:
The negation of the statement "An angle is a right angle if and only if it is of measure 90°" is:
"An angle is a right angle and it is not of measure 90° or an angle is of measure 90° and it is not a right angle."