Question

a) given lines z and y are parallel, determine the measure of each of the missing angles.

Explanation:

Step1: Vertical angles

Vertical angles are equal. So, $\angle1 = 74^{\circ}$ (vertical angle with the given $74^{\circ}$ angle).

Step2: Linear - pair

Since $\angle1+\angle3 = 180^{\circ}$ (linear - pair), $\angle3=180 - 74=106^{\circ}$. And $\angle2=\angle3 = 106^{\circ}$ (vertical angles).

Step3: Alternate - interior angles (for $39^{\circ}$ related angles)

$\angle12 = 39^{\circ}$ (vertical angle with the given $39^{\circ}$ angle). $\angle14=\angle12 = 39^{\circ}$ (vertical angles). $\angle13 = 180 - 39=141^{\circ}$ (linear - pair with $\angle12$), and $\angle13=\angle11 = 141^{\circ}$ (vertical angles).

Step4: Corresponding angles (for $y\parallel z$)

$\angle4=\angle12 = 39^{\circ}$ (corresponding angles as $y\parallel z$). $\angle6=\angle14 = 39^{\circ}$ (corresponding angles as $y\parallel z$).

Step5: Vertical angles (for $\angle5,\angle7,\angle9,\angle10,\angle15,\angle16,\angle17,\angle18$)

$\angle5=\angle4 = 39^{\circ}$ (vertical angles). $\angle7=\angle5 = 39^{\circ}$ (vertical angles). $\angle9=\angle3 = 106^{\circ}$ (vertical angles). $\angle10=\angle11 = 141^{\circ}$ (vertical angles). $\angle15=\angle9 = 106^{\circ}$ (vertical angles). $\angle16=\angle10 = 141^{\circ}$ (vertical angles). $\angle17=\angle15 = 106^{\circ}$ (vertical angles). $\angle18=\angle16 = 141^{\circ}$ (vertical angles).

Answer:

$\angle1 = 74^{\circ},\angle2 = 106^{\circ},\angle3 = 106^{\circ},\angle4 = 39^{\circ},\angle5 = 39^{\circ},\angle6 = 39^{\circ},\angle7 = 39^{\circ},\angle8 = 141^{\circ},\angle9 = 106^{\circ},\angle10 = 141^{\circ},\angle11 = 141^{\circ},\angle12 = 39^{\circ},\angle13 = 141^{\circ},\angle14 = 39^{\circ},\angle15 = 106^{\circ},\angle16 = 141^{\circ},\angle17 = 106^{\circ},\angle18 = 141^{\circ}$