Question

n, find the value of x.

Explanation:

Step1: Identify the relationship

Since lines \( m \) and \( n \) are parallel, and \( t \) is a transversal, the two angles \( (5x + 3)^\circ \) and \( (x + 3)^\circ \) are same - side interior angles? Wait, no, actually, looking at the diagram, if \( m\parallel n \), and the angles are formed by transversal \( t \), the two angles \( (5x + 3)^\circ \) and \( (x + 3)^\circ \) should be supplementary? Wait, no, maybe they are alternate - exterior or alternate - interior? Wait, no, let's re - examine. If \( m\parallel n \), and the transversal \( t \) cuts them, the angle \( (5x + 3)^\circ \) and \( (x + 3)^\circ \): actually, if we consider the linear pair or the fact that same - side interior angles are supplementary? Wait, no, maybe the two angles are supplementary? Wait, no, let's think again. Wait, the correct relationship: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but in the diagram, the two angles \( (5x + 3)^\circ \) and \( (x + 3)^\circ \): wait, maybe I made a mistake. Wait, actually, if \( m\parallel n \), and the transversal \( t \) intersects them, the angle \( (5x + 3)^\circ \) and \( (x + 3)^\circ \) are same - side interior angles? Wait, no, let's calculate. Wait, the sum of same - side interior angles is \( 180^\circ \). So \( (5x + 3)+(x + 3)=180 \).

Step2: Solve the equation

Combine like terms: \( 5x+x + 3 + 3=180 \)
Simplify: \( 6x+6 = 180 \)
Subtract 6 from both sides: \( 6x=180 - 6=174 \)
Divide both sides by 6: \( x=\frac{174}{6}=29 \)

Wait, but wait, maybe the angles are alternate - interior? No, if they were alternate - interior, they would be equal. Let's check: if \( 5x + 3=x + 3 \), then \( 4x = 0 \), \( x = 0 \), which doesn't make sense. So the correct relationship is that they are same - side interior angles, so supplementary. So the equation is \( (5x + 3)+(x + 3)=180 \)

Answer:

\( x = 29 \)