Question

in the figure below, n || o. find the values of z and x. z = \square x = \square

Explanation:

Step1: Determine z using vertical angles

Vertical angles are equal. The angle of \(60^\circ\) and \(z^\circ\) are vertical angles? Wait, no, actually, the angle adjacent to \(60^\circ\) and \(z\) - Wait, looking at the lines, \(n\) and \(o\) are parallel, and the transversal is line \(m\). Wait, actually, the angle \(z^\circ\) and the angle \(60^\circ\) - Wait, no, the angle \((3x - 9)^\circ\) and \(z^\circ\) are supplementary? Wait, no, first, the angle \(60^\circ\) and \((3x - 9)^\circ\) are corresponding angles because \(n \parallel o\) (assuming \(n\) and \(o\) are parallel, since they are both horizontal lines with the same transversal). So corresponding angles are equal. So \(3x - 9 = 60\)? Wait, no, wait: \(z\) and \(60^\circ\) - Wait, \(z\) and \((3x - 9)^\circ\) are supplementary? No, wait, let's re-examine.

Wait, the two horizontal lines are \(n\) (top) and \(o\) (bottom), with a transversal line \(m\). The angle at the bottom line \(o\) is \(60^\circ\), and the angle at the top line \(n\) adjacent to \((3x - 9)^\circ\) is \(z^\circ\). Wait, actually, \(z\) and \(60^\circ\) are corresponding angles? Wait, no, \(z\) and \((3x - 9)^\circ\) are supplementary (they form a linear pair), so \(z + (3x - 9) = 180\). But also, the angle \(60^\circ\) and \(z\) are equal? Wait, no, maybe the angle \(60^\circ\) and \((3x - 9)^\circ\) are equal because \(n \parallel o\) (corresponding angles). Let's check:

If \(n \parallel o\), then the corresponding angles are equal. So the angle at \(o\) (60°) and the angle at \(n\) (3x - 9)° are corresponding angles, so they should be equal. So:

\(3x - 9 = 60\)

Then, \(z\) and (3x - 9)° are supplementary (linear pair), so \(z + (3x - 9) = 180\). But if \(3x - 9 = 60\), then \(z = 180 - 60 = 120\)? Wait, no, that can't be. Wait, maybe I got the angles wrong.

Wait, let's look at the diagram again: the bottom line \(o\) has a \(60^\circ\) angle, and the top line \(n\) has \(z^\circ\) and \((3x - 9)^\circ\) forming a linear pair (so \(z + (3x - 9) = 180\)). Also, since \(n \parallel o\), the angle \(z^\circ\) and \(60^\circ\) are equal (alternate interior angles or corresponding angles). So \(z = 60\)? Wait, no, that would make \(z = 60\), then \(60 + (3x - 9) = 180\), so \(3x - 9 = 120\), so \(3x = 129\), \(x = 43\). But that doesn't seem right. Wait, maybe the angle \(60^\circ\) and \((3x - 9)^\circ\) are equal. Let's try that.

If \(3x - 9 = 60\), then solving for \(x\):

\(3x = 60 + 9 = 69\)

\(x = 69 / 3 = 23\)

Then, \(z\) and \((3x - 9)^\circ\) are supplementary, so \(z + 60 = 180\) (since \(3x - 9 = 60\)), so \(z = 180 - 60 = 120\). Wait, that makes sense. So:

Step1: Find x using corresponding angles

Since \(n \parallel o\) (assuming parallel lines), the corresponding angles are equal. So \(3x - 9 = 60\).

\(3x - 9 = 60\)

Step2: Solve for x

Add 9 to both sides:

\(3x = 60 + 9 = 69\)

Divide by 3:

\(x = \frac{69}{3} = 23\)

Step3: Find z using linear pair

\(z\) and \((3x - 9)^\circ\) form a linear pair, so they are supplementary (sum to \(180^\circ\)).

\(z + (3x - 9) = 180\)

We know \(3x - 9 = 60\), so:

\(z + 60 = 180\)

Step4: Solve for z

Subtract 60 from both sides:

\(z = 180 - 60 = 120\)

Answer:

\(z = 120\), \(x = 23\)