Question

solving algebraically for unknown values
find the values of x and y that make k || j and
m || n.
x =

y =

Explanation:

Step1: Use m || n (alternate interior angles)

Since \( m \parallel n \), the alternate interior angles \( (x + 50)^\circ \) and \( (x - 30)^\circ \) are supplementary? Wait, no, wait. Wait, when \( m \parallel n \) and the transversal is line \( k \), actually, since \( k \) is a transversal, and \( m \parallel n \), the same - side interior angles? Wait, no, let's look at the angles. The angle \( (x + 50)^\circ \) and \( (x - 30)^\circ \): when \( m \parallel n \), and the transversal is \( k \), these two angles should be equal? Wait, no, maybe I made a mistake. Wait, actually, when \( k \parallel j \) and \( m \parallel n \), let's re - analyze.

Wait, the correct approach: Since \( m \parallel n \), the consecutive interior angles (same - side interior angles) formed by transversal \( k \) should be supplementary. Wait, no, the angle \( (x + 50)^\circ \) and \( (x - 30)^\circ \): if \( m \parallel n \), then \( (x + 50)+(x - 30)=180 \)? Wait, no, that can't be. Wait, maybe they are alternate exterior angles? No, let's look at the diagram again.

Wait, actually, when \( m \parallel n \), the angle \( (x + 50)^\circ \) and \( (x - 30)^\circ \): since \( k \) is a transversal, and \( m \parallel n \), the two angles \( (x + 50)^\circ \) and \( (x - 30)^\circ \) are same - side interior angles? Wait, no, let's think about the parallel lines \( m \) and \( n \), cut by transversal \( k \). The angle \( (x + 50)^\circ \) and \( (x - 30)^\circ \): if we consider the direction of the lines, actually, \( (x + 50)=(x - 30)+180 \)? No, that doesn't make sense. Wait, I think I messed up. Let's start over.

Wait, the problem says \( k \parallel j \) and \( m \parallel n \). Let's first use \( m \parallel n \). The transversal is the line that intersects \( m \) and \( n \), which is line \( k \). So, the angles \( (x + 50)^\circ \) and \( (x - 30)^\circ \): when \( m \parallel n \), these two angles are same - side interior angles, so they should be supplementary. So, \( (x + 50)+(x - 30)=180 \).

Simplify the left - hand side: \( x + 50+x - 30 = 2x+20 \). So, \( 2x + 20=180 \).

Subtract 20 from both sides: \( 2x=180 - 20=160 \).

Divide both sides by 2: \( x = 80 \). Wait, no, that gives \( x=80 \), then \( x + 50=130 \), \( x - 30 = 50 \), and 130+50 = 180, which are supplementary. So that works for \( m \parallel n \).

Step2: Find y using k || j and m || n

Now that we have \( x = 80 \), the angle \( (x - 30)^\circ=(80 - 30)^\circ = 50^\circ \). Since \( k \parallel j \) and \( n \) is a transversal, the angle \( y \) and \( (x - 30)^\circ \) are corresponding angles (because \( m \parallel n \) and \( k \parallel j \), so the angle \( y \) should be equal to \( (x + 50)^\circ \)? Wait, no. Wait, when \( k \parallel j \) and \( n \) is a transversal, the angle \( y \) and the angle \( (x + 50)^\circ \) are corresponding angles? Wait, let's see. The line \( n \) is a transversal for \( k \) and \( j \). Since \( k \parallel j \), the corresponding angles are equal. The angle \( (x + 50)^\circ=(80 + 50)^\circ = 130^\circ \), so \( y = 130^\circ \)? Wait, no, wait. Wait, the angle \( (x - 30)^\circ = 50^\circ \), and since \( k \parallel j \), the angle \( y \) and \( (x + 50)^\circ \) are same - side interior angles? No, let's correct.

Wait, when \( k \parallel j \), and \( n \) is a transversal, the angle \( y \) and the angle \( (x + 50)^\circ \): since \( m \parallel n \), the angle \( (x + 50)^\circ \) and the angle \( y \): actually, the angle \( (x - 30)^\circ \) and \( y \): when \( k \parallel j \), the angle \( y \) and \( (x - 30)^\circ \) are supplementary? No, wait, let's use the value of \( x \).

We found \( x = 80 \), so \( x+50=130 \), \( x - 30 = 50 \). Now, since \( k \parallel j \), and \( n \) is a transversal, the angle \( y \) and the angle \( (x + 50)^\circ \) are corresponding angles (because the lines \( k \) and \( j \) are parallel, and \( n \) is a transversal), so \( y=(x + 50)^\circ=130^\circ \). Alternatively, since \( m \parallel n \), the angle \( (x + 50)^\circ \) and \( y \) are equal (corresponding angles) because \( k \parallel j \) and \( m \parallel n \).

Wait, let's re - check the first step. When \( m \parallel n \), the two angles \( (x + 50)^\circ \) and \( (x - 30)^\circ \) are same - side interior angles, so they are supplementary:

\( (x + 50)+(x - 30)=180 \)

\( 2x+20 = 180 \)

\( 2x=160 \)

\( x = 80 \)

Then, for \( y \): since \( k \parallel j \), and the transversal is \( n \), the angle \( y \) and \( (x + 50)^\circ \) are corresponding angles, so \( y=x + 50=80 + 50=130 \). Or, since \( m \parallel n \), the angle \( (x - 30)^\circ \) and \( y \) are supplementary (because \( k \parallel j \) and \( m \parallel n \)), \( 50 + y=180 \), so \( y = 130 \).

Answer:

\( x=\boxed{80} \), \( y=\boxed{130} \)