Question

find the value(s) of the variable(s). 23)

Explanation:

Step1: Identify angle relationship

The \(78^\circ\) angle and \((5x - 2)^\circ\) angle are complementary (they form a right angle with the vertical/horizontal lines, so their sum is \(90^\circ\))? Wait, no, actually, looking at the diagram, the two angles (78° and (5x - 2)°) and the right angle? Wait, no, the vertical and horizontal lines are perpendicular, so the angle between the horizontal (with 78°) and the line with (5x - 2)° should be such that 78° + (5x - 2)° = 90°? Wait, no, maybe they are alternate interior or something? Wait, no, the horizontal line and vertical line are perpendicular, so the angle between the horizontal (78° from vertical?) Wait, no, the diagram: a horizontal line and vertical line intersect, so the angles around the intersection. Wait, the 78° is between the horizontal (left) and vertical (up), and (5x - 2)° is between vertical (down) and horizontal (right). So those two angles (78° and (5x - 2)°) are equal? Wait, no, vertical angles? Wait, no, the horizontal line is straight, vertical line is straight. Wait, the angle of 78° and (5x - 2)°: since the vertical line is perpendicular to horizontal? No, wait, the horizontal and vertical lines are perpendicular, so the angle between horizontal (left) and vertical (up) is 78°, so the angle between horizontal (right) and vertical (down) should be equal? Wait, no, maybe 78° + (5x - 2)° = 90°? Wait, no, let's think again. The horizontal line and vertical line form a right angle (90°) between them. Wait, the 78° is one angle, and (5x - 2)° is another angle, such that 78° + (5x - 2)° = 90°? Wait, no, maybe they are equal because they are vertical angles or alternate angles. Wait, no, the 78° and (5x - 2)°: looking at the diagram, the horizontal line (left to right) and vertical line (up to down) intersect. The angle between the left horizontal and up vertical is 78°, and the angle between right horizontal and down vertical is (5x - 2)°. These two angles are equal because they are vertical angles? Wait, no, vertical angles are opposite each other. Wait, maybe the 78° and (5x - 2)° are complementary to the right angle? Wait, no, the horizontal and vertical lines are perpendicular, so the angle between them is 90°. So if one angle is 78°, the other angle (between horizontal and vertical on the other side) should be 90° - 78°? Wait, no, maybe the 78° and (5x - 2)° are equal. Wait, let's check the problem again. The diagram: a horizontal line (left and right arrows) and vertical line (up and down arrows) intersect. The angle between the left horizontal and up vertical is 78°, and the angle between right horizontal and down vertical is (5x - 2)°. So these two angles are equal because they are vertical angles? Wait, no, vertical angles are opposite. Wait, maybe the 78° and (5x - 2)° are equal. Wait, let's set up the equation. If the two angles are equal (since they are vertical angles or alternate angles), then 78 = 5x - 2? Wait, no, that would give 5x = 80, x=16. But wait, maybe they are complementary. Wait, no, the horizontal and vertical lines are perpendicular, so the angle between them is 90°. So 78° + (5x - 2)° = 90°? Let's check: 78 + (5x - 2) = 90 → 5x + 76 = 90 → 5x = 14 → x=2.8. But that seems odd. Wait, maybe the 78° and (5x - 2)° are equal because they are alternate interior angles or something. Wait, maybe the diagram is such that the 78° angle and (5x - 2)° angle are equal. Let's assume that. So 78 = 5x - 2. Then 5x = 80, so x=16. Wait, but let's think again. The horizontal line is straight, so the angle on the left (78°) and the angle on the right (5x - 2)°: if the vertical line is perpendicular, then the angle between horizontal (left) and vertical (up) is 78°, so the angle between horizontal (left) and vertical (down) is 90° - 78° = 12°, but that's not related. Wait, maybe the two angles (78° and (5x - 2)°) are equal because they are vertical angles. Wait, vertical angles are equal. So if the 78° angle and (5x - 2)° angle are vertical angles, then they are equal. So 78 = 5x - 2. Solving for x: 5x = 78 + 2 = 80 → x = 80 / 5 = 16. So that's the solution.

Step1: Set up the equation

Since the angles \(78^\circ\) and \((5x - 2)^\circ\) are equal (vertical angles), we set up the equation:
\(78 = 5x - 2\)

Step2: Solve for \(x\)

Add 2 to both sides:
\(78 + 2 = 5x\)
\(80 = 5x\)
Divide both sides by 5:
\(x = \frac{80}{5} = 16\)

Answer:

\(x = 16\)