Question

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

Explanation:

Step1: Identify supplementary angles

Since \( l \parallel m \), the angle of \( 131^\circ \) and \( (8x + 11)^\circ \) are supplementary (they form a linear pair). So, their sum is \( 180^\circ \).

$$ 131 + (8x + 11) = 180 $$

Step2: Simplify the equation

Combine like terms:

$$ 142 + 8x = 180 $$

Step3: Solve for \( x \)

Subtract 142 from both sides:

$$ 8x = 180 - 142 $$

$$ 8x = 38 $$

Divide both sides by 8:

$$ x = \frac{38}{8} = \frac{19}{4} = 4.75 $$

Wait, there's a mistake. Wait, actually, when lines are parallel, the consecutive interior angles are supplementary? Wait, no, looking at the diagram, the \( 131^\circ \) and \( (8x + 11)^\circ \) should be same - side interior angles? Wait, no, maybe I misread. Wait, actually, if \( l \parallel m \), the angle adjacent to \( 131^\circ \) (vertical angle or corresponding) – wait, no, let's re - examine.

Wait, the correct approach: The angle \( 131^\circ \) and \( (8x + 11)^\circ \) are same - side interior angles? No, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but maybe the \( 131^\circ \) and \( (8x + 11)^\circ \) are supplementary. Wait, let's do it again.

Correct Step1: Since \( l\parallel m \), the angle \( 131^\circ \) and \( (8x + 11)^\circ \) are supplementary (they add up to \( 180^\circ \)) because they are same - side interior angles. So:

$$ 131+(8x + 11)=180 $$

Step2: Combine like terms:

$$ 8x+142 = 180 $$

Step3: Subtract 142 from both sides:

$$ 8x=180 - 142=38 $$

Step4: Divide by 8:

$$ x=\frac{38}{8}=\frac{19}{4} = 4.75 $$

Wait, but that seems odd. Wait, maybe the angle \( (8x + 11)^\circ \) and \( 131^\circ \) are alternate interior angles? No, alternate interior angles are equal. Wait, maybe I made a mistake in the diagram interpretation. Wait, let's look again. The line \( l \) and \( m \) are parallel, cut by a transversal. The angle \( 131^\circ \) and \( (8x + 11)^\circ \): if the transversal cuts \( l \) and \( m \), then the \( 131^\circ \) and \( (8x + 11)^\circ \) – maybe the \( 131^\circ \) and \( (8x + 11)^\circ \) are supplementary. Wait, but let's check the calculation again.

Wait, \( 131+8x + 11 = 180\)
\(8x+142 = 180\)
\(8x=38\)
\(x = 4.75\) or \(x=\frac{19}{4}\)

But maybe the diagram is such that the angle \( (8x + 11)^\circ \) and \( 131^\circ \) are supplementary. So the calculation is correct.

Wait, no, maybe I messed up the angle relationship. Let's think differently. The angle \( 131^\circ \) and its adjacent angle (on line \( l \)) is \( 180 - 131=49^\circ \). Then, if \( l\parallel m \), the corresponding angle to \( 49^\circ \) would be \( (8x + 11)^\circ \)? Wait, that would be a different approach.

Ah! Here's the mistake. The \( 131^\circ \) angle and the angle adjacent to it (forming a linear pair) is \( 180 - 131 = 49^\circ \). Then, since \( l\parallel m \), the \( (8x + 11)^\circ \) angle is equal to \( 49^\circ \) (corresponding angles). So:

Step1: Find the supplementary angle of \( 131^\circ \)

$$ 180 - 131=49 $$

Step2: Set \( 8x + 11 \) equal to 49 (corresponding angles are equal when lines are parallel)

$$ 8x+11 = 49 $$

Step3: Solve for \( x \)
Subtract 11 from both sides:

$$ 8x=49 - 11 $$

$$ 8x = 38 $$

Wait, no, \( 49 - 11 = 38 \)? No, \( 49-11 = 38 \)? Wait, \( 49-11 = 38 \), then \( x=\frac{38}{8}=\frac{19}{4}=4.75 \). Wait, same result. So regardless of the approach, we get \( x = \frac{19}{4}=4.75 \) or \( x = 4.75 \).

Wait, maybe the initial assumption was correct. So the value of \( x \) is \( \frac{19}{4} \) or 4.75.

Answer:

\( x=\frac{19}{4} \) (or \( x = 4.75 \))