Question

given $m \parallel n$, find the value of $x$.

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \) and the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are same - side interior angles? Wait, no, actually, looking at the diagram, these two angles are supplementary? Wait, no, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary? Wait, no, in this case, the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are actually supplementary? Wait, no, let's re - examine. Wait, the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are adjacent and form a linear pair? Wait, no, actually, since \( m\parallel n \), and the transversal is the horizontal line. Wait, no, the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are supplementary? Wait, no, actually, if we look at the diagram, the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are same - side interior angles? Wait, no, let's think again. Wait, the sum of \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) should be \( 180^\circ \) because they are supplementary (linear pair or same - side interior angles when lines are parallel). So we set up the equation:
\( (4x - 4)+(6x - 6)=180 \)

Step2: Simplify the left - hand side of the equation

Combine like terms:
\( 4x+6x-4 - 6=180 \)
\( 10x-10 = 180 \)

Step3: Solve for x

Add 10 to both sides of the equation:
\( 10x-10 + 10=180 + 10 \)
\( 10x=190 \)? Wait, no, wait, that can't be right. Wait, maybe I made a mistake in the angle relationship. Wait, actually, the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are equal? Wait, no, if \( m\parallel n \), and the two angles are alternate interior angles? Wait, no, let's look at the diagram again. Wait, maybe the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are supplementary? Wait, no, maybe they are same - side interior angles. Wait, no, let's re - express the equation. Wait, maybe I messed up the angle relationship. Wait, actually, if we factor \( 4x - 4 = 4(x - 1) \) and \( 6x - 6=6(x - 1) \). Then \( 4(x - 1)+6(x - 1)=180 \), which is \( (4 + 6)(x - 1)=180 \), so \( 10(x - 1)=180 \). Then divide both sides by 10: \( x - 1 = 18 \), so \( x=19 \). Wait, that makes sense. Let's check:
If \( x = 19 \), then \( 4x-4=4\times19 - 4=76 - 4 = 72 \), and \( 6x - 6=6\times19-6 = 114 - 6 = 108 \). And \( 72+108 = 180 \), which is correct for supplementary angles.

So the correct steps are:

Step1: Set up the equation based on supplementary angles

Since the two angles \( (4x - 4)^\circ \) and \( (6x - 6)^\circ \) are supplementary (they form a linear pair or are same - side interior angles with \( m\parallel n \)), we have:
\( (4x - 4)+(6x - 6)=180 \)

Step2: Simplify the left - hand side

\( 4x+6x-4 - 6 = 180 \)
\( 10x-10=180 \)
Factor out 10 from the left - hand side:
\( 10(x - 1)=180 \)

Step3: Solve for x

Divide both sides by 10:
\( x - 1=\frac{180}{10}=18 \)
Add 1 to both sides:
\( x=18 + 1=19 \)

Answer:

\( x = 19 \)