Question

1. (6x + 19)° x°

Explanation:

Step1: Identify angle relationship

The two angles \( (6x + 19)^\circ \) and \( x^\circ \) are vertical angles? No, wait, looking at the diagram, actually, they seem to be supplementary? Wait, no, maybe they are adjacent and form a linear pair? Wait, no, the two angles here—wait, actually, the two angles \( (6x + 19)^\circ \) and \( x^\circ \) are vertical angles? Wait, no, maybe they are equal? Wait, no, maybe it's a case of vertical angles or maybe a linear pair? Wait, no, the diagram shows two intersecting lines, but there are three lines? Wait, no, two lines intersecting, and a third line? Wait, no, the diagram has three lines? Wait, no, maybe it's a case where the two angles \( (6x + 19)^\circ \) and \( x^\circ \) are supplementary? Wait, no, maybe they are equal? Wait, no, perhaps the two angles are vertical angles? Wait, no, maybe the angle \( (6x + 19)^\circ \) and \( x^\circ \) are adjacent and form a linear pair? Wait, no, let's re-examine. Wait, the diagram: two lines intersect, and there's a third line? Wait, no, maybe it's a case where the two angles \( (6x + 19)^\circ \) and \( x^\circ \) are equal? Wait, no, maybe it's a typo, but actually, the two angles are vertical angles? Wait, no, maybe they are supplementary? Wait, no, let's think again. Wait, the problem: when two lines intersect, vertical angles are equal, but here, maybe the two angles \( (6x + 19)^\circ \) and \( x^\circ \) are supplementary? Wait, no, maybe the diagram is of two parallel lines cut by a transversal? Wait, no, the diagram shows three lines? Wait, no, maybe it's a case where the angle \( (6x + 19)^\circ \) and \( x^\circ \) are adjacent and form a linear pair, so they add up to \( 180^\circ \)? Wait, no, that doesn't make sense. Wait, maybe the two angles are vertical angles? Wait, no, maybe the angle \( (6x + 19)^\circ \) is equal to \( x^\circ \)? No, that would give a negative x. Wait, maybe it's a case of alternate interior angles? Wait, no, the diagram: let's assume that the two angles \( (6x + 19)^\circ \) and \( x^\circ \) are supplementary? Wait, no, maybe the correct relationship is that \( 6x + 19 + x = 180 \)? Wait, no, that would be if they are supplementary. Wait, let's check:

Wait, maybe the two angles are adjacent and form a linear pair, so their sum is \( 180^\circ \). So:

\( 6x + 19 + x = 180 \)

Step2: Solve for x

Combine like terms:

\( 7x + 19 = 180 \)

Subtract 19 from both sides:

\( 7x = 180 - 19 \)

\( 7x = 161 \)

Divide both sides by 7:

\( x = \frac{161}{7} \)

\( x = 23 \)

Wait, but let's verify. If \( x = 23 \), then \( 6x + 19 = 6*23 + 19 = 138 + 19 = 157 \). Then 157 + 23 = 180, which is correct for a linear pair. So that makes sense.

Answer:

\( x = 23 \)