Question

in the figure below, k || h.
find the values of y and z.
y =
z =

Explanation:

Step1: Find y using vertical angles or corresponding angles

Since \( k \parallel h \), the angle \( y^\circ \) and the \( 81^\circ \) angle are corresponding angles (or vertical angles related), so \( y = 81 \)? Wait, no, wait. Wait, the angle \( (6z - 93)^\circ \) and \( y^\circ \) are supplementary? Wait, no, let's re-examine. Wait, the line \( j \) is a transversal. The angle \( (6z - 93)^\circ \) and \( y^\circ \) are adjacent, so they are supplementary? Wait, no, actually, since \( k \parallel h \), the angle \( y^\circ \) and the \( 81^\circ \) angle: wait, maybe \( (6z - 93)^\circ \) and \( 81^\circ \) are equal? Wait, because they are alternate interior angles? Wait, let's correct.

Wait, the angle \( (6z - 93)^\circ \) and the \( 81^\circ \) angle: since \( k \parallel h \), and \( j \) is transversal, so alternate interior angles are equal. So \( 6z - 93 = 81 \). Let's solve for \( z \):

\( 6z - 93 = 81 \)

\( 6z = 81 + 93 \)

\( 6z = 174 \)

\( z = 29 \)

Then, \( (6z - 93)^\circ = 6*29 - 93 = 174 - 93 = 81^\circ \). Then, \( (6z - 93)^\circ \) and \( y^\circ \) are supplementary (linear pair), so \( 81 + y = 180 \)? Wait, no, wait, the diagram: the angle \( (6z - 93)^\circ \) and \( y^\circ \) are adjacent, forming a linear pair? Wait, no, looking at the diagram: the horizontal line \( k \), with the transversal \( j \). The angle \( (6z - 93)^\circ \) is on the left, \( y^\circ \) is on the right, so they are adjacent, so they are supplementary? Wait, no, maybe \( y \) is equal to \( 81^\circ \)? Wait, I think I made a mistake. Wait, the angle \( y^\circ \) and the \( 81^\circ \) angle: since \( k \parallel h \), and \( j \) is transversal, \( y \) and \( 81^\circ \) are corresponding angles, so \( y = 81 \)? But then \( (6z - 93)^\circ \) and \( y \) are supplementary? Wait, no, let's re-express.

Wait, the angle \( (6z - 93)^\circ \) and \( y^\circ \) are adjacent, so they form a linear pair, so \( (6z - 93) + y = 180 \). But also, since \( k \parallel h \), the angle \( y^\circ \) and the \( 81^\circ \) angle: are they vertical angles? Wait, the \( 81^\circ \) angle is below, on line \( h \). So \( y^\circ \) and \( 81^\circ \) are vertical angles? Wait, no, vertical angles are opposite. Wait, maybe \( y \) is equal to \( 81^\circ \), and \( (6z - 93)^\circ \) is equal to \( 81^\circ \)? Wait, no, that would make \( (6z - 93) + y = 162 \), which is not 180. So my initial thought was wrong.

Wait, let's start over.

1. Find \( z \):

Since \( k \parallel h \), the angle \( (6z - 93)^\circ \) and the \( 81^\circ \) angle are alternate interior angles, so they are equal. So:

\( 6z - 93 = 81 \)

\( 6z = 81 + 93 \)

\( 6z = 174 \)

\( z = 29 \)

1. Find \( y \):

The angle \( (6z - 93)^\circ \) and \( y^\circ \) are supplementary (linear pair), so:

\( (6z - 93) + y = 180 \)

We know \( 6z - 93 = 81 \), so:

\( 81 + y = 180 \)

\( y = 180 - 81 \)

\( y = 99 \)

Wait, that makes sense. So:

• Solve for \( z \): \( 6z - 93 = 81 \) → \( z = 29 \)
• Solve for \( y \): \( y = 180 - (6z - 93) = 180 - 81 = 99 \)

Step1: Solve for z

Set alternate interior angles equal: \( 6z - 93 = 81 \)
\( 6z = 81 + 93 \)
\( 6z = 174 \)
\( z = 29 \)

Step2: Solve for y

\( (6z - 93)^\circ = 81^\circ \), so \( y = 180 - 81 = 99 \) (linear pair)

Answer:

\( y = 99 \), \( z = 29 \)