Exercise 12 Page 175

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

Let's start with the value of x.

The Value of x

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

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

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x+110=180

SubEqn

LHS-110=RHS-110

x=70

Thus, the value of x is 70^(∘). Now, we will place the value on the triangle to find the other values.

The Value of y

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

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y=110

As a result, the value of y is 110^(∘). Again, let's place it on the triangle and use it to find z.

The Value of z

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

z+40+110=180

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z+150=180

SubEqn

LHS-150=RHS-150

z=30

The value of the last variable is 30^(∘).