Sure, let's go through the process step-by-step.
Given that \( c \| d \), this means that lines \( c \) and \( d \) are parallel.
To prove: \( m_c = m_d \)
Statements
1. \( c \| d \)
2. Equations for lines \( c \) and \( d \): \( c = m_c x + b_c \) and \( d = m_d x + b_d \)
3. Set the equations equal because \( c \parallel d \) implies that the slopes are the same: \( m_c x + b_c = m_d x + b_d \)
4. Subtract \( m_d x \) from both sides to isolate the terms containing \( x \): \( m_c x - m_d x = b_d - b_c \)
5. Factor out \( x \): \( x(m_c - m_d) = b_d - b_c \)
6. Solve for \( x \): \( x = \frac{b_d - b_c}{m_c - m_d} \)
7. Since \( c \parallel d \), \( x \) does not exist for any \( x \) in the real number line, meaning the denominator must be zero: \( m_c - m_d = 0 \)
8. Combine similar terms: \( m_c = m_d \)
Reasons
1. Given (c \|d)
2. Write equations for lines \( c \) and \( d \)
3. Given that \( c \parallel d \)
4. Subtraction property of equality
5. Distributive property
6. Division property of equality
7. \( x \) does not exist; therefore, the denominator must be 0
8. Addition property of equality
Thus, since [tex]\( c \| d \)[/tex], we have proved that [tex]\( m_c = m_d \)[/tex].
