Question

diagram shows intersecting lines: a horizontal line (left and right arrows) and a line with upward and downward arrows intersecting at a point. one angle is ((13x + 19)^circ), another is (84^circ).

Explanation:

Step1: Identify vertical angles or linear pairs

The angle \((13x + 19)^\circ\) and \(84^\circ\) are vertical angles? Wait, no, actually, the angle \((13x + 19)^\circ\) and the angle adjacent to \(84^\circ\) on the straight line? Wait, no, looking at the diagram, the two angles \((13x + 19)^\circ\) and \(84^\circ\) are equal? Wait, no, actually, the angle \((13x + 19)^\circ\) and the angle that is vertical to \(84^\circ\)? Wait, no, let's think again. The two lines are intersecting, so vertical angles are equal. Wait, the angle \((13x + 19)^\circ\) and the angle opposite to it? Wait, no, the horizontal line and the vertical line (wait, no, the two lines are intersecting, forming vertical angles. Wait, the angle \((13x + 19)^\circ\) and the angle that is equal to \(84^\circ\)? Wait, no, actually, the angle \((13x + 19)^\circ\) and \(84^\circ\) are equal? Wait, no, maybe they are supplementary? Wait, no, let's check the diagram again. The horizontal line and the slanted line? Wait, no, the two lines are intersecting, so the angle \((13x + 19)^\circ\) and \(84^\circ\) are vertical angles? Wait, no, vertical angles are opposite each other. Wait, maybe the angle \((13x + 19)^\circ\) and \(84^\circ\) are equal because they are vertical angles. Wait, let's assume that \((13x + 19) = 84\)? No, that can't be, because \(13x + 19 = 84\) would give \(13x = 65\), \(x = 5\), but wait, maybe they are supplementary? Wait, no, if they are on a straight line, but no, the horizontal line is straight, so the angle \((13x + 19)^\circ\) and the angle adjacent to \(84^\circ\) on the straight line would be supplementary. Wait, no, the angle \(84^\circ\) and its vertical angle are equal, and the angle \((13x + 19)^\circ\) is equal to the vertical angle of \(84^\circ\)? Wait, no, let's look at the diagram again. The two lines intersect, so the angle \((13x + 19)^\circ\) and \(84^\circ\) are vertical angles, so they should be equal. Wait, so:

\(13x + 19 = 84\)? No, that would be if they are vertical angles, but wait, no, maybe the angle \((13x + 19)^\circ\) and \(84^\circ\) are supplementary? Wait, no, let's calculate. Wait, if the two lines are intersecting, then the angle \((13x + 19)^\circ\) and \(84^\circ\) are equal because they are vertical angles. Wait, let's check:

If \(13x + 19 = 84\), then \(13x = 84 - 19 = 65\), so \(x = 65 / 13 = 5\). Wait, that works. Wait, but maybe I made a mistake. Wait, no, vertical angles are equal. So the angle \((13x + 19)^\circ\) and \(84^\circ\) are vertical angles, so they are equal. So:

Step1: Set up the equation

Since the angles are vertical angles, they are equal. So:

\(13x + 19 = 84\)

Step2: Solve for x

Subtract 19 from both sides:

\(13x = 84 - 19\)

\(13x = 65\)

Divide both sides by 13:

\(x = \frac{65}{13}\)

\(x = 5\)

Answer:

\(x = 5\)