Question

given ( l parallel m parallel n ), find the value of ( x ).

Explanation:

Step1: Identify supplementary angles

Since \( l \parallel m \parallel n \), the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are supplementary (they form a linear pair or same - side interior angles related by parallel lines). So, \( (7x + 7)+133 = 180 \).

Step2: Simplify the equation

First, combine like terms: \( 7x+7 + 133=7x + 140 \). So the equation becomes \( 7x+140 = 180 \).

Step3: Solve for x

Subtract 140 from both sides: \( 7x=180 - 140=40 \)? Wait, no, wait. Wait, actually, if the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are supplementary, then \( 7x + 7+133 = 180 \) is wrong. Wait, actually, when lines are parallel, the angle \( (7x + 7)^\circ \) and the angle adjacent to \( 133^\circ \) (vertical or corresponding) - wait, no, let's re - examine. The angle \( (7x + 7)^\circ \) and \( 133^\circ \) should be equal? No, wait, no. Wait, if we look at the parallel lines, the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are same - side exterior angles? Wait, no, actually, the correct relationship is that \( 7x + 7\) and \( 133\) are supplementary? Wait, no, let's do it again.

Wait, the sum of two supplementary angles is \( 180^\circ \). If \( l\parallel m\parallel n \), then the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are supplementary. So:

\( 7x+7 + 133=180 \)

\( 7x+140 = 180 \)

Subtract 140 from both sides:

\( 7x=180 - 140 = 40 \)? No, that can't be. Wait, no, I made a mistake. Wait, actually, the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are equal? Wait, no, let's look at the diagram. The angle \( (7x + 7)^\circ \) and the angle that is supplementary to \( 133^\circ \) - wait, \( 180 - 133=47 \). No, that's not right. Wait, no, let's start over.

Since \( l\parallel m\parallel n \), the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are same - side interior angles? No, wait, the correct approach: the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are supplementary. So:

\( 7x+7 + 133 = 180 \) is incorrect. Wait, no, \( 7x + 7=180 - 133 \). Wait, yes! Because if two angles are supplementary, then \( A + B=180 \), so \( A = 180 - B \). So \( 7x+7=180 - 133 \).

Step4: Correct the equation

\( 7x+7 = 180 - 133 \)

\( 7x+7 = 47 \)

Subtract 7 from both sides: \( 7x=47 - 7 = 40 \)? No, that's not. Wait, I think I messed up the angle relationship. Let's look at the diagram again. The angle \( (7x + 7)^\circ \) and \( 133^\circ \) are actually equal? Wait, no, if the lines are parallel, the corresponding angles are equal. Wait, maybe the angle \( (7x + 7)^\circ \) and \( 133^\circ \) are equal. Wait, no, \( 7x+7 = 133 \)? Let's check: \( 7x=133 - 7=126 \), then \( x = 18 \). Ah, that makes sense. I see my mistake earlier. The angle \( (7x + 7)^\circ \) and \( 133^\circ \) are corresponding angles (because of the parallel lines), so they are equal. So the correct equation is \( 7x+7 = 133 \).

Step2 (corrected): Solve \( 7x+7 = 133 \)

Subtract 7 from both sides: \( 7x=133 - 7 = 126 \)

Step3 (corrected): Divide by 7

Divide both sides by 7: \( x=\frac{126}{7}=18 \)

Answer:

\( x = 18 \)