Question

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

Explanation:

Step1: Identify Alternate Interior Angles

Since \( j \parallel g \) and \( k \) is a transversal, the \( 64^\circ \) angle and \( z^\circ \) are alternate interior angles, so \( z = 64 \)? Wait, no, wait. Wait, the angle \( (3x + 29)^\circ \) and \( z^\circ \) are supplementary? Wait, no, let's re-examine. Wait, the \( 64^\circ \) angle and \( (3x + 29)^\circ \) – wait, no, \( j \parallel g \), so the angle with \( 64^\circ \) and \( z \): Wait, actually, \( z \) and \( 64^\circ \) – no, wait, the angle \( (3x + 29) \) and \( z \) are adjacent and form a linear pair? Wait, no, let's see: \( j \parallel g \), transversal \( k \). So the \( 64^\circ \) angle and \( (3x + 29)^\circ \) – wait, maybe \( z \) and \( 64^\circ \) are corresponding angles? Wait, no, let's correct. Wait, the angle marked \( 64^\circ \) and the angle \( z \) – no, wait, the angle \( (3x + 29) \) and \( z \) are vertical angles? No, wait, \( z \) and \( (3x + 29) \) are adjacent and supplementary? Wait, no, let's start over.

Wait, \( j \parallel g \), transversal \( k \). The angle of \( 64^\circ \) and the angle \( (3x + 29)^\circ \) – are they same-side interior angles? No, wait, maybe \( z \) is equal to \( 64^\circ \) because they are alternate interior angles? Wait, no, the angle \( z \) and \( (3x + 29) \) – wait, \( z + (3x + 29) = 180 \) (linear pair), and also, since \( j \parallel g \), the \( 64^\circ \) angle and \( (3x + 29) \) are same-side interior angles? No, same-side interior angles are supplementary. Wait, no, \( 64^\circ \) and \( (3x + 29) \) – wait, maybe \( 64^\circ \) and \( z \) are corresponding angles? Wait, no, let's look at the diagram again.

Wait, the line \( j \) and \( g \) are parallel, transversal \( k \). The angle with \( 64^\circ \) is on line \( g \), and \( z \) is on line \( j \), same position relative to transversal \( k \), so they are corresponding angles, so \( z = 64 \)? No, that can't be, because \( z \) and \( (3x + 29) \) are adjacent. Wait, no, \( z \) and \( (3x + 29) \) are supplementary (linear pair), so \( z + (3x + 29) = 180 \). Also, since \( j \parallel g \), the \( 64^\circ \) angle and \( (3x + 29) \) are same-side interior angles? No, same-side interior angles would be on the same side of transversal. Wait, maybe \( 64^\circ \) and \( (3x + 29) \) are equal because they are alternate interior angles? Wait, no, alternate interior angles are equal. Wait, if \( j \parallel g \), then the alternate interior angles are equal. So the angle of \( 64^\circ \) and \( (3x + 29)^\circ \) – are they alternate interior angles? Let's see: line \( j \) and \( g \), transversal \( k \). So the angle on \( j \) (above transversal) and angle on \( g \) (below transversal) – no, maybe \( 64^\circ \) and \( z \) are alternate interior angles, so \( z = 64 \), and then \( z + (3x + 29) = 180 \), so \( 64 + 3x + 29 = 180 \), so \( 3x + 93 = 180 \), \( 3x = 87 \), \( x = 29 \). Wait, that makes sense.

Step1: Determine Relationship for \( z \)

Since \( j \parallel g \) and \( k \) is a transversal, the \( 64^\circ \) angle and \( z^\circ \) are alternate interior angles. Thus, \( z = 64 \).

Step2: Solve for \( x \)

The angles \( z^\circ \) and \( (3x + 29)^\circ \) form a linear pair (supplementary), so:

$$ z + (3x + 29) = 180 $$

Substitute \( z = 64 \):

$$ 64 + 3x + 29 = 180 $$

Simplify:

$$ 3x + 93 = 180 $$

Subtract 93 from both sides:

$$ 3x = 180 - 93 = 87 $$

Divide by 3:

$$ x = \frac{87}{3} = 29 $$

Answer:

\( z = 64 \), \( x = 29 \)