Question

question
given: ( c parallel d )
prove: ( m_c = m_d )
select the correct answer from each drop - down menu.

statements\treasons
( c parallel d )\tgiven
( c = m_cx + b_c )\twrite equations for lines ( c ) and ( d )
( d = m_dx + b_d )\t
( m_cx + b_c = m_dx + b_d )\tsolve for ( x )
( m_cx - m_dx = b_d - b_c )\tsubtraction property of equality
( x(m_c - m_d) = b_d - b_c )\tdistributive property
( x=\frac{b_d - b_c}{m_c - m_d} )\tdivision property of equality
( m_c - m_d = 0 )\tx does not exist; therefore, the denominator must be 0
( m_c = m_d )\taddition property of equality

Explanation:

Step1: Identify the task

We need to prove that the slopes of parallel lines \( c \) and \( d \) are equal (\( m_c = m_d \)) by matching statements with their corresponding reasons.

Step2: Analyze the first statement-reason pair

• Statement: \( c \parallel d \)
• Reason: This is given in the problem, so the reason is "given".

Step3: Analyze the second statement-reason pair

• Statement: \( c: y = m_cx + b_c \), \( d: y = m_dx + b_d \)
• Reason: To represent the equations of lines \( c \) and \( d \) in slope - intercept form (\( y=mx + b \)), we "write equations for lines \( c \) and \( d \)".

Step4: Analyze the third statement-reason pair

• Statement: \( m_cx + b_c=m_dx + b_d \)
• Reason: Since parallel lines (non - vertical) have the same \( y \) - value for the same \( x \) - value (they never intersect), we set their equations equal. This is part of "solving for \( x \)" (finding the \( x \) - value where they would intersect, but since they are parallel, we use this equation to analyze the slopes and intercepts).

Step5: Analyze the fourth statement-reason pair

• Statement: \( m_cx - m_dx=b_d - b_c \)
• Reason: We subtract \( m_dx \) and \( b_c \) from both sides of the equation \( m_cx + b_c=m_dx + b_d \). This is the "subtraction property of equality" (if \( a = b \), then \( a - c=b - c \)).

Step6: Analyze the fifth statement-reason pair

• Statement: \( x(m_c - m_d)=b_d - b_c \)
• Reason: We factor out \( x \) from the left - hand side of the equation \( m_cx - m_dx=b_d - b_c \). This is the "distributive property" ( \( ax - bx=x(a - b) \)).

Step7: Analyze the sixth statement-reason pair

• Statement: \( x=\frac{b_d - b_c}{m_c - m_d} \)
• Reason: We divide both sides of the equation \( x(m_c - m_d)=b_d - b_c \) by \( (m_c - m_d) \) to solve for \( x \). But for the equation \( x=\frac{b_d - b_c}{m_c - m_d} \) to be defined, \( m_c - m_d

eq0 \) (otherwise, if \( m_c - m_d = 0\), the denominator is zero and \( x \) is undefined). However, since we are dealing with parallel lines (non - vertical), and we know that if \( m_c=m_d \), the lines are parallel (and non - intersecting), we can also think about the fact that if \( m_c
eq m_d \), the lines would intersect, which contradicts \( c\parallel d \). But from the equation \( x=\frac{b_d - b_c}{m_c - m_d} \), if \( m_c - m_d = 0\), \( x \) is undefined (since division by zero is not allowed), which means our assumption that \( m_c
eq m_d \) is wrong. So we conclude \( m_c - m_d = 0\).

• The act of dividing both sides of the equation \( x(m_c - m_d)=b_d - b_c \) by \( (m_c - m_d) \) (assuming \( m_c - m_d

eq0 \) at first) is the "division property of equality" (if \( a=b\) and \( c
eq0\), then \( \frac{a}{c}=\frac{b}{c} \)).

Step8: Analyze the seventh statement-reason pair

• Statement: \( m_c - m_d = 0\)
• Reason: Since \( x=\frac{b_d - b_c}{m_c - m_d} \) and \( x \) must be a real number (or undefined, but for parallel lines, the only way for \( x \) to be undefined is when the denominator \( m_c - m_d = 0\) because if \( m_c - m_d

eq0\), \( x \) is defined and the lines would intersect, which is not possible for parallel lines). So we say " \( x \) does not exist; therefore, the denominator must be 0".

Step9: Analyze the eighth statement-reason pair

• Statement: \( m_c=m_d \)
• Reason: From \( m_c - m_d = 0\), we add \( m_d \) to both sides of the equation (using the "addition property of equality": if \( a - b=0\), then \( a=b\) when we add \( b \) to both sides).

Answer:

The correct matching of statements and reasons is as follows:

1. \( c\parallel d \) - given
2. \( c:y = m_cx + b_c\), \( d:y = m_dx + b_d \) - write equations for lines \( c \) and \( d \)
3. \( m_cx + b_c=m_dx + b_d \) - solve for \( x \)
4. \( m_cx - m_dx=b_d - b_c \) - subtraction property of equality
5. \( x(m_c - m_d)=b_d - b_c \) - distributive property
6. \( x=\frac{b_d - b_c}{m_c - m_d} \) - division property of equality
7. \( m_c - m_d = 0\) - \( x \) does not exist; therefore, the denominator must be 0
8. \( m_c=m_d \) - addition property of equality

(If we are just to state the final conclusion of the proof, the answer is \( m_c = m_d \) as we have proven that the slopes of parallel lines \( c \) and \( d \) are equal.)