Question

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

Explanation:

Step1: Find the value of \( z \)

Since \( m\parallel n \), \( z \) and \( 78^{\circ} \) are alternate - interior angles.
By the alternate - interior angles theorem, \( z = 78 \).

Step2: Find the value of \( y \)

Since \( p\parallel q \), \( z^{\circ} \) and \( (6y - 102)^{\circ} \) are same - side interior angles.
By the same - side interior angles theorem, \( z+(6y - 102)=180 \).
Substitute \( z = 78 \) into the equation: \( 78+(6y - 102)=180 \).
Simplify the left - hand side: \( 6y-24 = 180 \).
Add 24 to both sides: \( 6y=180 + 24\), so \( 6y=204 \).
Divide both sides by 6: \( y=\frac{204}{6}=34 \).

Answer:

\( z = 78\), \( y = 34\)