Question

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

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \) and \( t \) is a transversal, the angles \( (8x - 5)^\circ \) and \( (x + 5)^\circ \) are same - side interior angles? Wait, no, actually, looking at the diagram, the angle \( (8x - 5)^\circ \) and the angle adjacent to \( (x + 5)^\circ \) (vertical angles or supplementary? Wait, no, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, the angle \( (8x - 5)^\circ \) and \( (x + 5)^\circ \) are actually same - side interior angles? Wait, no, let's re - examine. If we consider the linear pair and the parallel lines, the angle \( (8x - 5)^\circ \) and \( (x + 5)^\circ \) should be supplementary? Wait, no, actually, when \( m\parallel n \), the angle \( (8x - 5)^\circ \) and the angle that is vertical to the angle adjacent to \( (x + 5)^\circ \) are equal, but maybe a better approach: the angle \( (8x - 5)^\circ \) and \( (x + 5)^\circ \) are same - side interior angles? Wait, no, let's think again. The correct relationship is that \( (8x - 5)^\circ+(x + 5)^\circ = 180^\circ \)? Wait, no, that would be if they are same - side interior angles. Wait, no, actually, looking at the diagram, the angle \( (8x - 5)^\circ \) and the angle \( (x + 5)^\circ \) are same - side interior angles? Wait, no, maybe alternate interior angles? Wait, no, let's see: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, let's check the positions. The upper line is \( m \), lower is \( n \), transversal \( t \). The angle \( (8x - 5)^\circ \) is on the upper line, above the transversal, and the angle \( (x + 5)^\circ \) is on the lower line, below the transversal, on the same side of the transversal. So they are same - side interior angles, so they should be supplementary. So:
\( (8x - 5)+(x + 5)=180 \)

Step2: Solve the equation

Simplify the left - hand side of the equation:
\( 8x-5 + x + 5=180 \)
Combine like terms: \( 8x+x-5 + 5=180\), which simplifies to \( 9x=180 \)
Divide both sides by 9: \( x=\frac{180}{9}=20 \)

Answer:

\( x = 20 \)