Question

1. if ( j parallel k ), find the value of ( x ).

Explanation:

Step1: Identify angle relationship

Since \( j \parallel k \), the two angles \( (2x - 8)^\circ \) and \( (9x - 10)^\circ \) are alternate interior angles (or corresponding, depending on the diagram's exact configuration, but they should be equal as lines are parallel). So we set them equal:
\( 2x - 8 = 9x - 10 \)

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides:
\( -8 = 7x - 10 \)

Add 10 to both sides:
\( 2 = 7x \)

Divide both sides by 7:
\( x = \frac{2}{7} \)? Wait, no, wait—maybe they are same - side or supplementary? Wait, maybe I misidentified. Wait, looking at the diagram, maybe they are alternate exterior or actually, maybe vertical angles? Wait, no, let's re - examine. Wait, maybe the angles are equal as alternate interior. Wait, no, maybe I made a mistake. Wait, let's check again. Wait, if \( j \parallel k \), and the transversal, maybe the two angles are equal. Wait, no, maybe they are supplementary? Wait, no, let's do the equation again. Wait, maybe the angles are equal. Wait, \( 2x - 8=9x - 10 \) gives \( - 7x=-2 \), so \( x = \frac{2}{7} \), which seems odd. Wait, maybe they are same - side interior angles? Then they should be supplementary. So \( (2x - 8)+(9x - 10)=180 \). Let's try that.

Step3: Correct angle relationship (supplementary)

If the angles are same - side interior angles (since lines are parallel, same - side interior angles are supplementary), then:
\( (2x - 8)+(9x - 10)=180 \)

Combine like terms:
\( 11x-18 = 180 \)

Add 18 to both sides:
\( 11x=198 \)

Divide by 11:
\( x = 18 \)

Ah, that makes sense. So the correct relationship is supplementary (same - side interior angles), so we set their sum to 180.

Answer:

\( x = 18 \)