Question

find the value of z.

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So $y = 18^{\circ}$ since the angle opposite to the $18^{\circ}$ angle and $y$ are vertical angles.

Step2: Use angle - sum property of a circle

The sum of angles around a point is $360^{\circ}$. We know two angles are $64^{\circ}$ and $18^{\circ}$, and $y = 18^{\circ}$. Let's find $z$.
$z+64^{\circ}+18^{\circ}+y + x=360^{\circ}$. Also, since the pair of angles with measures $x$ and $64^{\circ}$ are vertical angles, $x = 64^{\circ}$.
Substitute $x = 64^{\circ}$ and $y = 18^{\circ}$ into the equation:
$z+64^{\circ}+18^{\circ}+18^{\circ}+64^{\circ}=360^{\circ}$.
$z+(64 + 18+18 + 64)^{\circ}=360^{\circ}$.
$z + 164^{\circ}=360^{\circ}$.

Step3: Solve for $z$

Subtract $164^{\circ}$ from both sides of the equation:
$z=360^{\circ}-164^{\circ}$.
$z = 196^{\circ}$.

Answer:

$196^{\circ}$