Question

find the values of the missing angles in the diagram below.
show your work here
w =
x =
y =
z =

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. The angle vertical to the $76^{\circ}$ angle has a measure of $76^{\circ}$.

Step2: Find $x$ using angle - sum property of a triangle

In the triangle with angles $x$, $76^{\circ}$, and the angle adjacent to the $100^{\circ}$ angle. The adjacent angle to the $100^{\circ}$ angle is $180 - 100=80^{\circ}$ (linear - pair property). By the angle - sum property of a triangle ($x + 76+80 = 180$), we get $x=180-(76 + 80)=24^{\circ}$.

Step3: Find $y$ using angle - sum property of a triangle

In the triangle with angles $y$, $27^{\circ}$, and the angle vertical to the $76^{\circ}$ angle (which is $76^{\circ}$). By the angle - sum property of a triangle ($y+27 + 76=180$), we get $y=180-(27 + 76)=77^{\circ}$.

Step4: Find $z$ using linear - pair property

The angle adjacent to $z$ and the $27^{\circ}$ angle and the $y = 77^{\circ}$ angle form a straight - line. So the angle adjacent to $z$ is $180-(27 + 77)=76^{\circ}$. Then, since $z$ and this angle are vertical angles, $z = 76^{\circ}$.

Answer:

$x = 24^{\circ}$, $y = 77^{\circ}$, $z = 76^{\circ}$