Question

l || m || n, find the value of x.

Explanation:

Step1: Identify angle relationship

Since \( l \parallel m \parallel n \), the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles? Wait, no, actually, when dealing with parallel lines cut by a transversal, if we consider the alternate - interior or corresponding angles, but here, since the lines are parallel, the angle \( (2x + 20)^\circ \) and \( 54^\circ \) should be supplementary? Wait, no, let's correct. Wait, actually, when three parallel lines are cut by a transversal, the consecutive interior angles (same - side interior angles) are supplementary. Wait, let's look at the diagram. The angle \( (2x + 20)^\circ \) and the angle adjacent to \( 54^\circ \) (the vertical angle or corresponding angle) should have a relationship. Wait, actually, the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles? No, wait, let's think again. If \( l\parallel m\parallel n \), and the transversal cuts them, then the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles, so they should be supplementary? Wait, no, same - side interior angles are supplementary when lines are parallel. Wait, but maybe I made a mistake. Wait, actually, the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles, so \( (2x + 20)+54 = 180 \)? Wait, no, that can't be. Wait, maybe it's a typo in the problem, and the angle is \( (2x + 20)^\circ \) and the angle corresponding to \( 54^\circ \) is such that they are same - side interior angles. Wait, let's start over.

Wait, the correct approach: When two parallel lines are cut by a transversal, same - side interior angles are supplementary. Since \( l\parallel m\parallel n \), the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles, so:

\( (2x + 20)+54=180 \)

Step2: Solve for x

First, simplify the left - hand side of the equation:

\( 2x+20 + 54=2x + 74 \)

So the equation becomes \( 2x+74 = 180 \)

Subtract 74 from both sides:

\( 2x=180 - 74 \)

\( 2x = 106 \)

Divide both sides by 2:

\( x=\frac{106}{2}=53 \)

Wait, that doesn't seem right. Wait, maybe the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are corresponding angles? If they are corresponding angles, then \( 2x + 20=54 \), but that would give \( 2x=34 \), \( x = 17 \), which also doesn't seem right. Wait, maybe there is a mistake in the problem statement. Wait, looking at the original problem, maybe the angle is \( (2x + 20)^\circ \) and the angle is equal to \( 54^\circ \) plus some? Wait, no, let's check the problem again.

Wait, maybe the angle \( (2x + 20)^\circ \) and \( 54^\circ \) are same - side interior angles, so they should be supplementary. So:

\( 2x+20 + 54=180 \)

\( 2x+74 = 180 \)

\( 2x=180 - 74=106 \)

\( x = 53 \)

But let's verify. If \( x = 53 \), then \( 2x+20=2\times53 + 20=106 + 20 = 126 \). And \( 126+54 = 180 \), which satisfies the supplementary angle condition for same - side interior angles when lines are parallel. So that makes sense.

Answer:

\( x = 53 \)