Write the negatin of each of the statement :
Two lines are parallel if and only if they have the same slope.

Answer

To find the negation of the statement "Two lines are parallel if and only if they have the same slope," we need to break down the statement and understand its logical structure.

### Step 1: Identify the components of the statement
The statement can be expressed in logical terms as:
- Let P: "Two lines are parallel."
- Let Q: "They have the same slope."

The original statement can be written as:
- P if and only if Q (P ↔ Q)

### Step 2: Understand the meaning of "if and only if"
The phrase "if and only if" means that both conditions must either be true or false together. In logical terms, this can be expressed as:
- (P → Q) ∧ (Q → P)

### Step 3: Write the negation of the statement
To negate the statement P ↔ Q, we can use the logical equivalence:
- The negation of (P ↔ Q) is (P ∧ ¬Q) ∨ (¬P ∧ Q).

This means:
- "Two lines are parallel and do not have the same slope" or "Two lines do not have the same slope and are parallel."

### Step 4: Final negated statement
Thus, the negation of the original statement is:
- "Two lines are parallel and do not have the same slope, or two lines do not have the same slope and are parallel."

### Summary of the negation
The negation of the statement "Two lines are parallel if and only if they have the same slope" is:
- "Two lines are parallel and do not have the same slope, or two lines do not have the same slope and are parallel."

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