Question

b. find the value of the variable.
(3x + 2)°
50°

Explanation:

Step1: Identify vertical angles

Vertical angles are equal, so \(3x + 2 = 50\) (wait, no, actually, the angle \(50^\circ\) and \((3x + 2)^\circ\) – wait, no, looking at the diagram, the \(50^\circ\) angle and \((3x + 2)^\circ\) angle: wait, no, maybe they are vertical angles? Wait, no, let's check again. Wait, the two angles: the \(50^\circ\) and \((3x + 2)^\circ\) – are they vertical angles? Wait, vertical angles are opposite each other when two lines intersect. So if two lines intersect, the opposite angles are equal. So if one angle is \(50^\circ\), the opposite angle (which is \((3x + 2)^\circ\)) should be equal? Wait, no, wait, maybe I made a mistake. Wait, no, let's see: the diagram shows two intersecting lines, with one angle \(50^\circ\) and the other \((3x + 2)^\circ\) as vertical angles? Wait, no, maybe they are adjacent? Wait, no, vertical angles are equal. So if the angle is \(50^\circ\), then the vertical angle is also \(50^\circ\)? Wait, no, the angle given is \((3x + 2)^\circ\) and \(50^\circ\) – maybe they are vertical angles? Wait, no, maybe I misread. Wait, the problem is to find \(x\) where the angle is \((3x + 2)^\circ\) and the vertical angle is \(50^\circ\)? Wait, no, that can't be. Wait, no, maybe the two angles are vertical angles, so \(3x + 2 = 50\)? Wait, no, that would give \(3x = 48\), \(x = 16\). Wait, let's check again.

Wait, actually, when two lines intersect, vertical angles are equal. So if one angle is \(50^\circ\), the angle opposite to it (which is \((3x + 2)^\circ\)) should be equal to \(50^\circ\)? Wait, no, that would mean \(3x + 2 = 50\). Let's solve that.

Step2: Solve for \(x\)

Set \(3x + 2 = 50\) (since vertical angles are equal). Subtract 2 from both sides: \(3x = 50 - 2 = 48\). Then divide by 3: \(x = \frac{48}{3} = 16\). Wait, but wait, is that correct? Wait, maybe I made a mistake in identifying the angles. Wait, maybe the angles are supplementary? No, vertical angles are equal. Let me confirm: vertical angles are formed by two intersecting lines, and they are opposite each other, so they have the same measure. So if one angle is \(50^\circ\), the opposite angle (which is \((3x + 2)^\circ\)) is also \(50^\circ\)? Wait, no, that would mean \(3x + 2 = 50\), so \(x = 16\). Let's check: \(3(16) + 2 = 48 + 2 = 50\), which matches. So that's correct.

Answer:

\(x = 16\)