Question

in the figure below, m || n and p || q. find the values of z and x, z = 115 x = \boxed{} (6x - 83)° 65° z°

Explanation:

Step1: Identify Alternate Interior Angles

Since \( m \parallel n \) and \( p \parallel q \), the angle \( z^\circ \) and \( 65^\circ \) are supplementary (linear pair), but we already know \( z = 115 \). Now, for \( x \), the angle \( x^\circ \) and \( (6x - 83)^\circ \) are alternate interior angles (because \( m \parallel n \) and \( q \) is a transversal), so they should be equal. Wait, actually, also, since \( p \parallel q \), the \( 65^\circ \) and \( x^\circ \) are supplementary? Wait, no, let's re-examine. Wait, \( z = 115 \) is correct because \( 65 + z = 180 \) (linear pair). Now, for \( x \), since \( m \parallel n \) and the transversal \( q \), the angle \( x \) and \( (6x - 83) \) should be equal? Wait, no, maybe corresponding angles. Wait, actually, since \( p \parallel q \), the angle \( 65^\circ \) and the angle equal to \( (6x - 83)^\circ \) (because of \( m \parallel n \))? Wait, let's use the fact that \( x \) and \( (6x - 83) \) are equal? Wait, no, let's see: since \( m \parallel n \), the angle \( x \) and the angle \( (6x - 83) \) are alternate interior angles, so \( x = 6x - 83 \)? No, that can't be. Wait, maybe \( z = 115 \), and since \( p \parallel q \), the angle \( x \) and \( 65^\circ \) are supplementary? No, \( z = 115 \) is supplementary to \( 65 \). Wait, maybe the angle \( (6x - 83) \) is equal to \( z \)? Wait, \( z = 115 \), so \( 6x - 83 = 115 \)? Wait, no, let's check the diagram again. The angle \( x \) and \( (6x - 83) \): since \( m \parallel n \), and \( q \) is a transversal, so \( x = 6x - 83 \)? No, that would give negative. Wait, maybe \( x \) and \( 65^\circ \) are equal? No, \( z = 115 \) is adjacent to \( 65 \). Wait, let's start over.

Given \( m \parallel n \) and \( p \parallel q \). First, \( 65^\circ \) and \( z^\circ \) are linear pair, so \( 65 + z = 180 \), so \( z = 115 \) (correct as given). Now, for \( x \): since \( p \parallel q \), the angle \( x \) and the angle \( (6x - 83) \) are corresponding angles? Wait, no, \( m \parallel n \), so the angle \( x \) and the angle \( (6x - 83) \) are equal? Wait, no, let's use the fact that \( z = 115 \), and since \( p \parallel q \), the angle \( x \) and \( 65^\circ \) are supplementary? No, \( z = 115 \), and \( x \) and \( 65^\circ \): wait, maybe \( x = 65 \)? No, that doesn't fit. Wait, maybe the angle \( (6x - 83) \) is equal to \( z = 115 \)? So \( 6x - 83 = 115 \). Let's solve that: \( 6x = 115 + 83 = 198 \), so \( x = 198 / 6 = 33 \). Wait, that makes sense. So \( 6x - 83 = 6*33 - 83 = 198 - 83 = 115 \), which is equal to \( z = 115 \). So that's correct. So the equation is \( 6x - 83 = z \), and \( z = 115 \), so \( 6x - 83 = 115 \).

Step2: Solve for \( x \)

Set up the equation: \( 6x - 83 = 115 \)
Add 83 to both sides: \( 6x = 115 + 83 = 198 \)
Divide both sides by 6: \( x = \frac{198}{6} = 33 \)

Answer:

\( x = 33 \)