Question

given ( m parallel n ), find the value of ( x ).
answer attempt 1 out of 2
( x = )

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \), the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are same - side interior angles? Wait, no, actually, looking at the diagram, they are supplementary? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, these two angles look like they are adjacent and form a linear pair? Wait, no, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are actually alternate interior angles? Wait, no, let's re - examine. Wait, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \): since \( m\parallel n \), and the transversal is the horizontal line. Wait, actually, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are supplementary? No, wait, no. Wait, when two parallel lines are cut by a transversal, consecutive interior angles are supplementary, but also, if we look at the vertical angles or alternate interior angles. Wait, no, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are actually equal? Wait, no, that can't be. Wait, maybe they are same - side interior angles? Wait, no, let's think again. Wait, the correct relationship: when two parallel lines are cut by a transversal, the same - side interior angles are supplementary. But in this case, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are actually supplementary? Wait, no, wait, the sum of same - side interior angles is \( 180^\circ \). Wait, no, maybe I made a mistake. Wait, looking at the diagram, the two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \) are adjacent and form a linear pair? No, they are on the same side of the transversal. Wait, actually, the correct relationship is that \( (8x - 1)+(9x - 11)=180 \)? No, that would be if they are same - side interior angles. Wait, no, wait, maybe they are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, let's check the diagram again. The two angles \( (8x - 1)^\circ \) and \( (9x - 11)^\circ \): since \( m\parallel n \), and the transversal is the horizontal line, the two angles are actually equal? Wait, no, that doesn't make sense. Wait, maybe I got the angle types wrong. Wait, let's start over.

When two parallel lines are cut by a transversal, alternate interior angles are equal. Let's see, the angle \( (8x - 1)^\circ \) and the angle \( (9x - 11)^\circ \): if we consider the transversal, maybe they are alternate interior angles. So we set \( 8x-1 = 9x - 11 \).

Step2: Solve for x

Start with the equation \( 8x-1=9x - 11 \).

Subtract \( 8x \) from both sides: \( 8x-1 - 8x=9x - 11-8x \), which simplifies to \( - 1=x - 11 \).

Then add 11 to both sides: \( -1 + 11=x-11 + 11 \), so \( x = 10 \).

Wait, let's check. If \( x = 10 \), then \( 8x-1=8\times10 - 1=79 \), and \( 9x - 11=9\times10-11 = 79 \). So they are equal, which means they are alternate interior angles, so that makes sense because \( m\parallel n \), alternate interior angles are equal.

Answer:

\( x = 10 \)