Question

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

Explanation:

Step1: Use vertical - angles property

Vertical angles are equal. So, $x = 70^{\circ}$ since the angle opposite to the $70^{\circ}$ angle is vertical to it.

Step2: Use linear - pair property

A linear pair of angles sums to $180^{\circ}$. For the angle adjacent to the $91^{\circ}$ angle, let's call it $y_1$. Then $y_1=180 - 91=89^{\circ}$.

Step3: Use angle - sum property of a triangle

In the triangle formed, we know one angle is $y_1 = 89^{\circ}$ and another is $x = 70^{\circ}$. Let the third angle be $z$. Using the fact that the sum of angles in a triangle is $180^{\circ}$, we have $z=180-(89 + 70)=21^{\circ}$.

Step4: Use vertical - angles property for $w$

The angle vertical to the $42^{\circ}$ angle and $w$ are related. Since vertical angles are equal and considering the straight - line with the $42^{\circ}$ angle, we know that $w = 42^{\circ}$.

Answer:

$w = 42^{\circ}$, $x = 70^{\circ}$, $y = 89^{\circ}$, $z = 21^{\circ}$