Let's find the value of each variable for the given triangle.

Let's start with the value of $x .$

In order to find $x,$ we will use the Triangle Angle-Sum Theorem.

Triangle Angle-Sum Theorem
The sum of the measures of the angles of a triangle is 180.

We can illustrate this theorem on the following diagram.

m ∠ A + m ∠ B + m ∠ C = 180

Let's apply the theorem using the measurements of the small triangle on the left.

x + 30 + 80 = 180

Add terms

x + 110 = 180

$LHS - 110 = RHS - 110$

x = 70

Thus, the value of

x

is

7 0^{\circ} .

Now, we will place the value on the triangle to find the other values.

Next, we will find $y .$ To do that, we will use the Triangle Exterior Angle Theorem.

Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

We can illustrate this theorem as shown below.

m ∠ 1 = m ∠ 2 + m ∠ 3

Applying this theorem once again to the small triangle on the left, we can find the value of

y .

y = 30 + 80

Add terms

y = 110

As a result, the value of

y

is

11 0^{\circ} .

Again, let's place it on the triangle and use it to find

z .

We will use the Triangle Angle-Sum Theorem one more time to find the value of

z .

z + 40 + 110 = 180

Add terms

z + 150 = 180

$LHS - 150 = RHS - 150$

z = 30

The value of the last variable is

3 0^{\circ} .

Exercise