Question

in the figure below, m || n and p || q. find the values of x and y. x = \square y = \square

Explanation:

Step1: Find x using corresponding angles

Since \( p \parallel q \) and \( m \parallel n \), the angle \( x \) and the \( 64^\circ \) angle are corresponding angles (or alternate interior angles, depending on the transversal). So \( x = 64^\circ \)? Wait, no, wait. Wait, the angle \( 64^\circ \) and the angle \( x \): Wait, actually, since \( p \parallel q \) and the transversal is the line cutting through \( m \) and \( n \), wait, no, let's re-examine. Wait, the angle \( (3y - 38)^\circ \) and \( x \): since \( m \parallel n \), the angle \( (3y - 38)^\circ \) and \( x \) are alternate interior angles? Wait, no, maybe first, the \( 64^\circ \) angle and the angle \( (3y - 38)^\circ \): since \( p \parallel q \), the angle \( 64^\circ \) and \( (3y - 38)^\circ \) are corresponding angles? Wait, no, let's see: \( m \parallel n \), and \( p \) and \( q \) are parallel. So the angle \( 64^\circ \) and the angle \( (3y - 38)^\circ \): since \( p \parallel q \), the transversal is the line that's not \( p \) or \( q \), wait, maybe the angle \( x \) and the \( 64^\circ \) angle: no, \( x \) is formed by line \( p \) and line \( n \). Wait, actually, since \( m \parallel n \), the angle \( (3y - 38)^\circ \) and \( x \) are alternate interior angles, so they are equal? Wait, no, maybe the \( 64^\circ \) angle and \( x \): no, \( x \) is adjacent to the angle formed by line \( p \) and \( n \). Wait, let's start over.

First, since \( p \parallel q \), the angle \( 64^\circ \) and the angle \( (3y - 38)^\circ \) are corresponding angles (because \( m \parallel n \) and \( p \parallel q \), so the figure is a parallelogram, so opposite angles are equal). Wait, no, maybe the angle \( x \) and the \( 64^\circ \) angle: since \( p \) is a transversal cutting \( m \) and \( n \), and \( m \parallel n \), the angle \( x \) and the angle \( (3y - 38)^\circ \) are alternate interior angles, so they are equal. But also, since \( p \parallel q \), the angle \( 64^\circ \) and \( (3y - 38)^\circ \) are equal? Wait, maybe:

Since \( p \parallel q \), the angle \( 64^\circ \) and \( (3y - 38)^\circ \) are corresponding angles (because \( m \parallel n \) is the other pair of parallel lines), so \( 3y - 38 = 64 \). Let's solve for \( y \):

\( 3y - 38 = 64 \)

\( 3y = 64 + 38 \)

\( 3y = 102 \)

\( y = 34 \)

Then, since \( m \parallel n \), the angle \( x \) and the angle \( (3y - 38)^\circ \) are alternate interior angles, so \( x = 3y - 38 \). Wait, but we just found \( 3y - 38 = 64 \), so \( x = 64 \)? Wait, no, that can't be. Wait, maybe \( x \) is equal to \( 64^\circ \)? Wait, no, maybe the angle \( x \) and the \( 64^\circ \) angle are supplementary? No, that doesn't make sense. Wait, let's check the diagram again. The angle \( 64^\circ \) is on line \( n \), and \( x \) is on line \( n \) formed by line \( p \). So line \( p \) intersects line \( n \) at \( x \), and line \( q \) intersects line \( n \) at \( 64^\circ \). Since \( p \parallel q \), the angle \( x \) and \( 64^\circ \) are corresponding angles? Wait, no, if \( p \parallel q \), then the angle \( x \) and \( 64^\circ \) should be equal? Wait, no, maybe \( x \) is equal to \( 64^\circ \), and \( 3y - 38 = x \). Wait, that would make sense. So:

Since \( p \parallel q \), the angle \( x \) and \( 64^\circ \) are corresponding angles, so \( x = 64 \).

Then, since \( m \parallel n \), the angle \( (3y - 38)^\circ \) and \( x \) are alternate interior angles, so \( 3y - 38 = x \).

We know \( x = 64 \), so:

\( 3y - 38 = 64 \)

\( 3y = 64 + 38 \)

\( 3y = 102 \)

\( y = 34 \)

Yes, that makes sense. So \( x = 64 \) and \( y = 34 \).

Step1: Determine x using parallel lines

Since \( p \parallel q \), the angle \( x \) and the \( 64^\circ \) angle are corresponding angles (because \( m \parallel n \) and \( p, q \) are parallel). Thus, \( x = 64 \).

Step2: Solve for y using parallel lines

Since \( m \parallel n \), the angle \( (3y - 38)^\circ \) and \( x \) (which is \( 64^\circ \)) are alternate interior angles, so \( 3y - 38 = 64 \).

Solve for \( y \):

\( 3y = 64 + 38 \)

\( 3y = 102 \)

\( y = \frac{102}{3} = 34 \)

Answer:

\( x = 64 \)

\( y = 34 \)