Question

in the figure below, k || l and m || n. find the values of z and y.
(5y + 5)° on line k, z° and 75° on line l, with lines m and n intersecting them.
z =
y =

Explanation:

Step1: Find z using parallel lines and transversal

Since \( m \parallel n \) and \( l \) is a transversal, consecutive interior angles are supplementary? Wait, no, \( z \) and \( 75^\circ \): Wait, \( m \parallel n \), so alternate interior angles? Wait, \( k \parallel l \), \( m \) is a transversal? Wait, no, \( m \parallel n \), and \( l \) is a transversal. Wait, the angle \( z \) and \( 75^\circ \): since \( m \parallel n \), and \( l \) is a transversal, \( z \) and \( 75^\circ \) are same - side interior angles? No, wait, \( z \) and \( 75^\circ \): actually, since \( m \parallel n \), and \( l \) is a transversal, \( z + 75^\circ=180^\circ\)? Wait, no, that's if they are same - side interior. Wait, no, looking at the diagram, \( z \) and \( 75^\circ \): since \( m \parallel n \), and \( l \) is a transversal, \( z \) and \( 75^\circ \) are supplementary? Wait, no, maybe \( z = 180 - 75=105\)? Wait, no, wait \( k \parallel l \), and \( m \) is a transversal. Wait, the angle \( (5y + 5)^\circ \) and \( z^\circ \): since \( k \parallel l \), \( (5y + 5)^\circ=z^\circ \) (corresponding angles). And since \( m \parallel n \), \( z^\circ+75^\circ = 180^\circ \) (same - side interior angles). So first, find \( z \):

\( z + 75=180 \)

\( z=180 - 75 = 105 \)

Step2: Find y using the corresponding angles

Since \( k \parallel l \), the angle \( (5y + 5)^\circ \) and \( z^\circ \) are corresponding angles, so \( 5y+5 = z \). We know \( z = 105 \), so:

\( 5y+5 = 105 \)

Subtract 5 from both sides: \( 5y=105 - 5=100 \)

Divide both sides by 5: \( y=\frac{100}{5}=20 \)

Answer:

\( z = 105 \), \( y = 20 \)