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 can follow these steps:

### Step 1: Identify the components of the statement
The statement can be broken down into two parts:
- Let \( p \): "Two lines are parallel."
- Let \( q \): "They have the same slope."

### Step 2: Understand the structure of the statement
The statement is of the form "p if and only if q," which can be written as:
\[ p \iff q \]
This means that both \( p \) implies \( q \) and \( q \) implies \( p \).

### Step 3: Write the negation of the statement
The negation of a biconditional statement \( p \iff q \) is given by:
\[ \neg (p \iff q) \]
This can be expressed as:
\[ \neg p \iff \neg q \]
This means that "not p" is equivalent to "not q."

### Step 4: Determine the negations of \( p \) and \( q \)
- The negation of \( p \) ("Two lines are parallel") is "Two lines are not parallel."
- The negation of \( q \) ("They have the same slope") is "They do not have the same slope."

### Step 5: Combine the negations
Now we can write the negation of the original statement:
\[ \text{"They do not have the same slope if and only if two lines are not parallel."} \]

### Final Statement
Thus, the negation of the statement "Two lines are parallel if and only if they have the same slope" is:
"They do not have the same slope if and only if two lines are not parallel."

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