Question

given ( l parallel m parallel n ), find the value of ( x ).
image of three parallel lines ( l ), ( m ), ( n ) cut by a transversal, with angle ( 129^circ ) and ( (2x + 11)^circ ) at the intersection with ( n )
answer attempt 1 out of 2
( x = ) input box submit answer

Explanation:

Step1: Identify the relationship

Since \( l \parallel m \parallel n \), the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles or supplementary? Wait, no, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary. But here, looking at the diagram, \( (2x + 11)^\circ \) and \( 129^\circ \) should be supplementary? Wait, no, actually, if we consider the parallel lines \( m \) and \( n \) cut by the transversal, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles, so they should add up to \( 180^\circ \)? Wait, no, wait. Wait, maybe I made a mistake. Wait, actually, if the lines are parallel, and the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are adjacent and form a linear pair? Wait, no, the diagram shows that \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles? Wait, no, let's think again.

Wait, when three parallel lines \( l\), \( m\), \( n \) are cut by a transversal, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are supplementary? Wait, no, actually, if we look at the position, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles, so they should be supplementary. So we can set up the equation:

\( (2x + 11)+129 = 180 \)

Step2: Solve the equation

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

\( 2x+11 + 129=2x + 140 \)

So the equation becomes \( 2x+140 = 180 \)

Subtract 140 from both sides:

\( 2x=180 - 140 \)

\( 2x = 40 \)

Divide both sides by 2:

\( x=\frac{40}{2}=20 \)

Wait, wait, that can't be right. Wait, maybe I messed up the relationship. Wait, maybe the angle \( (2x + 11)^\circ \) is equal to \( 129^\circ \)? No, that would not make sense. Wait, no, maybe the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are alternate interior angles? No, alternate interior angles are equal. Wait, maybe I made a mistake in the diagram interpretation.

Wait, let's re - examine. If \( l\parallel m\parallel n \), and the transversal cuts them, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles, so they are supplementary. So:

\( 2x+11 + 129=180 \)

\( 2x+140 = 180 \)

\( 2x=180 - 140 \)

\( 2x = 40 \)

\( x = 20 \)? Wait, no, wait, maybe the angle \( (2x + 11)^\circ \) is equal to \( 129^\circ \)? No, that would give \( 2x+11 = 129 \), then \( 2x=129 - 11=118 \), \( x = 59 \). Wait, I think I made a mistake in the angle relationship.

Wait, let's look at the diagram again. The angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles? No, actually, if the lines are parallel, and the transversal cuts them, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are supplementary? Wait, no, maybe they are equal? Wait, no, let's think about the properties of parallel lines.

Wait, when two parallel lines are cut by a transversal, alternate interior angles are equal, corresponding angles are equal, and same - side interior angles are supplementary.

Looking at the diagram, the angle \( (2x + 11)^\circ \) and \( 129^\circ \) are same - side interior angles, so they should add up to \( 180^\circ \). Wait, but if we solve \( 2x + 11+129 = 180 \), we get \( 2x=40 \), \( x = 20 \). But let's check again.

Wait, maybe the angle \( (2x + 11)^\circ \) is equal to \( 129^\circ \)? No, that would be if they are alternate interior angles. Wait, maybe the diagram is such that \( (2x + 11)^\circ \) and \( 129^\circ \) are supplementary. Wait, let's re - do the calculation.

Equation: \( 2x+11 + 129=180 \)

Combine like terms: \( 2x+140 = 180 \)

Subtract 140 from both sides: \( 2x=180 - 140=40 \)

Divide by 2: \( x = 20 \). Wait, but let's check the angle: \( 2x+11=2\times20 + 11 = 51 \), and \( 51+129 = 180 \), which is correct for supplementary angles. So that works.

Wait, I think I was confused earlier. So the correct equation is \( (2x + 11)+129 = 180 \) because they are same - side interior angles (supplementary) when the lines are parallel.

Answer:

\( x = 20 \)