To prove that [tex]\( m_c = m_d \)[/tex], let's carefully go through the statements and the corresponding reasons step-by-step. Here are the detailed explanations:
1. Statement: [tex]\( c \parallel d \)[/tex]
- Reason: Given
Since [tex]\( c \parallel d \)[/tex] (lines [tex]\( c \)[/tex] and [tex]\( d \)[/tex] are parallel), we start by acknowledging the given information.
2. Statement: [tex]\( c = m_c x + b_c \)[/tex]
- Reason: Write equations for lines [tex]\( c \)[/tex] and [tex]\( d \)[/tex]
The equation of line [tex]\( c \)[/tex] is represented as [tex]\( c = m_c x + b_c \)[/tex], where [tex]\( m_c \)[/tex] is the slope of line [tex]\( c \)[/tex], and [tex]\( b_c \)[/tex] is the y-intercept of line [tex]\( c \)[/tex].
3. Statement: [tex]\( d = m_d x + b_d \)[/tex]
- Reason: Write equations for lines [tex]\( c \)[/tex] and [tex]\( d \)[/tex]
The equation of line [tex]\( d \)[/tex] is similarly represented as [tex]\( d = m_d x + b_d \)[/tex], where [tex]\( m_d \)[/tex] is the slope of line [tex]\( d \)[/tex], and [tex]\( b_d \)[/tex] is the y-intercept of line [tex]\( d \)[/tex].
4. Statement: [tex]\( m_c x + b_c = m_d x + b_d \)[/tex]
- Reason: Since the lines are parallel and must remain distinct if they have different intercepts, this statement helps start the derivation to compare their forms. If they were coincident, they would overlap entirely, but drawing from the context that we assume typically parallel lines don't overlap completely, we balance their forms.
5. Statement: [tex]\( m_c x - m_d x = b_d - b_c \)[/tex]
- Reason: Rearrange the terms to solve for [tex]\( x \)[/tex]
After rearranging and combining like terms, we proceed with:
[tex]\[
m_c x - m_d x = b_d - b_c
\][/tex]
Factoring out [tex]\( x \)[/tex]:
[tex]\[
(m_c - m_d) x = b_d - b_c
\][/tex]
6. Statement: Solve for [tex]\( x \)[/tex]
- Reason: Subtraction property of equality
If the slopes [tex]\( m_c \)[/tex] and [tex]\( m_d \)[/tex] were different, then [tex]\( x \)[/tex] would have a specific value for which the above equality holds. However, since the lines are parallel, their slopes must be the same to satisfy the equation for all [tex]\( x \)[/tex] values. Therefore:
[tex]\[
m_c - m_d = 0
\][/tex]
Which simplifies to:
[tex]\[
m_c = m_d
\][/tex]
Therefore, we have proven that the slopes [tex]\( m_c \)[/tex] and [tex]\( m_d \)[/tex] are indeed equal given the lines [tex]\( c \)[/tex] and [tex]\( d \)[/tex] are parallel.
