Question

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

Explanation:

Step1: Identify the angle relationship

Since \( m \parallel n \) and the transversal \( t \) intersects them, the two angles \( (2x - 2)^\circ \) and \( (2x - 10)^\circ \) are same - side interior angles? Wait, no, looking at the diagram, actually, when two parallel lines are cut by a transversal, the alternate - exterior or alternate - interior angles? Wait, no, let's re - examine. Wait, the angle \( (2x - 2)^\circ \) and the angle adjacent to \( (2x - 10)^\circ \) (vertical angles or corresponding angles)? Wait, no, actually, since \( m\parallel n \), the angle \( (2x - 2)^\circ \) and \( (2x - 10)^\circ \) are supplementary? Wait, no, maybe they are alternate - interior angles? Wait, no, let's think again. Wait, if we consider the vertical angle of \( (2x - 10)^\circ \), it should be equal to \( (2x - 2)^\circ \) if they are corresponding angles? Wait, no, maybe I made a mistake. Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, alternate - interior angles are equal, corresponding angles are equal.

Wait, looking at the diagram, the angle \( (2x - 2)^\circ \) and the angle that is vertical to \( (2x - 10)^\circ \) (let's call it \( \angle A \)): since \( m\parallel n \), \( (2x - 2)^\circ \) and \( \angle A \) are same - side interior angles? No, wait, maybe the two angles \( (2x - 2)^\circ \) and \( (2x - 10)^\circ \) are supplementary? Wait, no, let's check the correct relationship. Wait, actually, when two parallel lines are cut by a transversal, the consecutive interior angles are supplementary. But in this case, if we look at the positions, the angle \( (2x - 2)^\circ \) and the angle \( (2x - 10)^\circ \) are actually same - side interior angles? Wait, no, maybe they are alternate - exterior angles? Wait, no, let's do it step by step.

Wait, the correct approach: when \( m\parallel n \), and the transversal \( t \) cuts them, the angle \( (2x - 2)^\circ \) and the angle \( (2x - 10)^\circ \) are supplementary? Wait, no, that can't be. Wait, maybe they are equal? Wait, no, let's think about the diagram again. Wait, maybe the angle \( (2x - 2)^\circ \) and the angle \( (2x - 10)^\circ \) are same - side interior angles, so they should be supplementary. Wait, no, same - side interior angles sum to \( 180^\circ \). Wait, but let's check:

Wait, actually, the two angles \( (2x - 2)^\circ \) and \( (2x - 10)^\circ \) are same - side interior angles? Wait, no, maybe I got the diagram wrong. Wait, the line \( m \) is above, \( n \) is below, transversal \( t \) is slanting. The angle on \( m \) is \( (2x - 2)^\circ \), and the angle on \( n \) is \( (2x - 10)^\circ \) on the same side of the transversal? Wait, no, maybe they are alternate - interior angles. Wait, alternate - interior angles are equal. So if \( m\parallel n \), then \( 2x - 2=2x - 10 \)? That can't be, because \( - 2 eq - 10 \). So that's wrong.

Wait, maybe the angle \( (2x - 10)^\circ \) and the vertical angle of \( (2x - 2)^\circ \) are supplementary. Wait, the vertical angle of \( (2x - 2)^\circ \) is also \( (2x - 2)^\circ \). Then, if the angle \( (2x - 2)^\circ \) and \( (2x - 10)^\circ \) are same - side interior angles, then \( (2x - 2)+(2x - 10) = 180 \). Let's try that.

Step2: Set up the equation

If the two angles are same - side interior angles (since \( m\parallel n \)), then their sum is \( 180^\circ \). So:

\( (2x - 2)+(2x - 10)=180 \)

Step3: Simplify the left - hand side

Combine like terms:

\( 2x+2x-2 - 10 = 180 \)

\( 4x-12 = 180 \)

Step4: Solve for \( x \)

Add 12 to both sides:

\( 4x=180 + 12 \)

\( 4x=192 \)

Divide both sides by 4:

\( x=\frac{192}{4}=48 \)

Answer:

\( x = 48 \)