Question

in the figure below, ( m parallel n ) and ( p parallel q ). find the values of ( y ) and ( z ).

(figure with lines ( m, n ) (horizontal, parallel), ( p, q ) (slanted, parallel); ( q ) intersects ( m ) at ( 72^circ ), ( p ) intersects ( n ) at ( (3z + 3)^circ ), ( q ) intersects ( n ) at ( y^circ ).)

( y = square )
( z = square )

Explanation:

Step1: Find y using corresponding angles

Since \( m \parallel n \) and \( q \) is a transversal, the \( 72^\circ \) angle and \( y^\circ \) are corresponding angles. Corresponding angles are equal when lines are parallel. So \( y = 72 \).

Step2: Find z using corresponding angles

Since \( p \parallel q \) and \( n \) is a transversal, the \( (3z + 3)^\circ \) angle and the angle equal to \( 72^\circ \) (corresponding to \( m \)) are corresponding angles. Wait, actually, since \( m \parallel n \) and \( p \parallel q \), the \( (3z + 3)^\circ \) should equal \( 72^\circ \)? Wait no, let's correct. Wait, \( m \parallel n \), so the angle adjacent to \( 72^\circ \) (supplementary) or wait, no. Wait, \( p \parallel q \), so the angle \( (3z + 3)^\circ \) and the angle that is equal to \( 72^\circ \) (because \( m \parallel n \))? Wait, no, let's use the fact that \( p \parallel q \), so the angle \( (3z + 3)^\circ \) and the angle corresponding to \( 72^\circ \) (since \( m \parallel n \)) are equal. Wait, actually, since \( m \parallel n \), the angle at \( m \) (72°) and the angle at \( n \) (y) are equal (corresponding), so \( y = 72 \). Then, since \( p \parallel q \), the angle \( (3z + 3)^\circ \) and the angle \( y^\circ \) are corresponding angles? Wait, no, the lines \( p \) and \( q \) are parallel, and \( n \) is a transversal. Wait, the angle \( (3z + 3)^\circ \) and the angle that is equal to \( 72^\circ \) (because \( m \parallel n \)): actually, since \( m \parallel n \), the angle above \( n \) (corresponding to 72°) is 72°, and since \( p \parallel q \), the angle \( (3z + 3)^\circ \) is equal to that 72°? Wait, no, let's set \( 3z + 3 = 72 \)? Wait, no, that would be if they are corresponding. Wait, solving \( 3z + 3 = 72 \): \( 3z = 72 - 3 = 69 \), so \( z = 23 \). Wait, let's check: \( 3(23) + 3 = 69 + 3 = 72 \), which matches. So yes, because \( p \parallel q \), the \( (3z + 3)^\circ \) angle and the angle equal to \( 72^\circ \) (from \( m \parallel n \)) are corresponding angles, so they are equal. Thus, \( 3z + 3 = 72 \). Solving:
\( 3z = 72 - 3 = 69 \)
\( z = \frac{69}{3} = 23 \)

Answer:

\( y = 72 \)
\( z = 23 \)