Question

given the figure below, find the values of x and z.
the figure shows two intersecting straight lines, forming angles. one angle is (118^circ), another angle is ((15x - 88)^circ), and the third angle is (z^circ).

Explanation:

Step1: Identify vertical angles

Vertical angles are equal, so \((15x - 88)^\circ = 118^\circ\)

Step2: Solve for \(x\)

\[

$$\begin{align*} 15x - 88 &= 118\\ 15x &= 118 + 88\\ 15x &= 206\\ x &= \frac{206}{15}\\ x &= 13.733... \end{align*}$$

\]
Wait, that can't be right. Wait, maybe the angle \((15x - 88)\) and \(118^\circ\) are supplementary? Wait, no, looking at the diagram, maybe the angle \((15x - 88)\) and \(118^\circ\) are vertical angles? Wait, no, maybe I made a mistake. Wait, actually, when two lines intersect, adjacent angles are supplementary. Wait, no, let's re-examine.

Wait, the angle \(118^\circ\) and the angle \((15x - 88)^\circ\) are vertical angles? No, maybe they are supplementary? Wait, no, vertical angles are equal. Wait, maybe the angle \(z\) and \(118^\circ\) are supplementary, and \((15x - 88)\) and \(z\) are vertical angles? Wait, no, let's start over.

Wait, the diagram shows two intersecting lines. So, the angle \(118^\circ\) and the angle \((15x - 88)^\circ\) are vertical angles? Wait, no, vertical angles are opposite each other. Wait, maybe the angle \(118^\circ\) and the angle \((15x - 88)^\circ\) are equal? Wait, no, that would give \(15x - 88 = 118\), so \(15x = 206\), \(x = 13.733\), which is not an integer. Maybe I misread the angle. Wait, the angle is \(118^\circ\), and the other angle is \((15x - 88)^\circ\). Wait, maybe they are supplementary? Wait, no, adjacent angles formed by intersecting lines are supplementary. Wait, no, vertical angles are equal, and adjacent angles are supplementary.

Wait, let's look again. The angle \(118^\circ\) and the angle \(z\) are adjacent, so they are supplementary. So \(z + 118 = 180\), so \(z = 180 - 118 = 62\). Then, the angle \((15x - 88)^\circ\) is vertical to \(z\), so \(15x - 88 = z = 62\). Then, \(15x = 62 + 88 = 150\), so \(x = 10\). Ah, that makes sense. I made a mistake earlier by thinking \((15x - 88)\) is vertical to \(118\), but actually, \(z\) is vertical to \((15x - 88)\), and \(z\) is supplementary to \(118\).

So, correcting:

Step1: Find \(z\)

Since \(z\) and \(118^\circ\) are supplementary (adjacent angles on a straight line), we have:
\[
z + 118 = 180
\]
\[
z = 180 - 118 = 62
\]

Step2: Find \(x\)

Since \((15x - 88)^\circ\) and \(z^\circ\) are vertical angles, they are equal:
\[
15x - 88 = z
\]
Substitute \(z = 62\):
\[
15x - 88 = 62
\]
\[
15x = 62 + 88
\]
\[
15x = 150
\]
\[
x = \frac{150}{15} = 10
\]

Answer:

\(x = 10\), \(z = 62\)