Question

solve for the value of v.
(9v - 3)° 84°
answer attempt 1 out of 2
v =

Explanation:

Step1: Identify angle relationship

The two angles \((9v - 3)^\circ\) and \(84^\circ\) are complementary (they form a right angle, so their sum is \(90^\circ\)). So, we set up the equation: \((9v - 3)+84 = 90\).

Step2: Simplify the equation

First, combine like terms: \(9v+81 = 90\). Then, subtract 81 from both sides: \(9v=90 - 81\), which simplifies to \(9v = 9\).

Step3: Solve for \(v\)

Divide both sides by 9: \(v=\frac{9}{9}=1\). Wait, no, wait. Wait, actually, looking at the diagram again, maybe the two angles and the right angle? Wait, no, the vertical line and the horizontal line: the angle between the vertical and the right - pointing horizontal is \(84^\circ\), and the angle between the vertical and the left - pointing horizontal is \((9v - 3)^\circ\). So those two angles should add up to \(90^\circ\)? Wait, no, maybe I made a mistake. Wait, actually, the vertical line and the horizontal line (the two horizontal rays) form a straight line? No, the vertical line is perpendicular to the horizontal? Wait, no, the diagram: there is a vertical ray, a right - pointing horizontal ray, and a left - pointing horizontal ray. The angle between the vertical ray and the right - pointing horizontal ray is \(84^\circ\), and the angle between the vertical ray and the left - pointing horizontal ray is \((9v - 3)^\circ\). So those two angles should add up to \(90^\circ\)? Wait, no, a right angle is \(90^\circ\), so \((9v - 3)+84 = 90\). Let's re - do the steps:

1. Start with the equation: \(9v-3 + 84=90\)
2. Combine the constants: \(9v + 81=90\)
3. Subtract 81 from both sides: \(9v=90 - 81=9\)
4. Divide both sides by 9: \(v = 1\)? Wait, that seems wrong. Wait, maybe the two angles and the right angle? No, wait, maybe the sum of \((9v - 3)\) and \(84\) is \(90\) because they are complementary (since the vertical line is perpendicular to the horizontal line, forming a right angle). Wait, but let's check again. If \(v = 1\), then \(9v-3=9(1)-3 = 6\), and \(6 + 84=90\), which is correct. Wait, but that seems too simple. Wait, maybe I misinterpreted the diagram. Wait, maybe the two angles are supplementary? No, because there is a vertical line. Wait, the vertical line and the horizontal line (the two horizontal rays) form a straight line? No, the vertical line is perpendicular to the horizontal, so the angle between the two horizontal rays is \(180^\circ\), and the vertical line splits it into two angles: one is \(84^\circ\) and the other is \((9v - 3)^\circ\), and since the vertical line is perpendicular to the horizontal? No, if the vertical line is perpendicular to the horizontal, then the angle between vertical and horizontal is \(90^\circ\). So the angle between vertical and right - horizontal is \(84^\circ\), so the angle between vertical and left - horizontal is \(90 - 84=6^\circ\). So \(9v-3 = 6\), then \(9v=9\), \(v = 1\). Wait, that's correct.

Wait, no, let's re - examine the diagram. The vertical ray, the right - pointing horizontal ray, and the left - pointing horizontal ray. The angle between vertical and right - horizontal is \(84^\circ\), angle between vertical and left - horizontal is \((9v - 3)^\circ\). Since vertical is perpendicular to horizontal (assuming the vertical and horizontal are perpendicular, forming a right angle), then \((9v - 3)+84 = 90\). So:

\(9v-3+84 = 90\)

\(9v + 81=90\)

\(9v=90 - 81\)

\(9v = 9\)

\(v = 1\). Wait, but that seems correct. Wait, maybe I made a mistake in the angle relationship. Let's think again. If the vertical line is perpendicular to the horizontal line (the two horizontal rays are a straight line, and the vertical line is perpendicular to the horizontal line), then the angle between vertical and right - horizontal is \(84^\circ\), so the angle between vertical and left - horizontal is \(90 - 84 = 6^\circ\). So \(9v-3=6\), so \(9v=9\), \(v = 1\). Yes, that's correct.

Wait, no, wait a second. Maybe the two angles \((9v - 3)\) and \(84\) are supplementary? No, because the vertical line and the horizontal line: if the horizontal line is a straight line, and the vertical line is coming out of the intersection, then the angle between vertical and right - horizontal is \(84^\circ\), angle between vertical and left - horizontal is \((9v - 3)^\circ\), and the sum of these two angles should be \(90^\circ\) (since vertical is perpendicular to horizontal). So the equation is correct.

Answer:

\(v = 1\)