Exercise 51 Page 238

From the figure we can make out a triangle where an exterior angle and two nonadjacent angles are labeled. Let's highlight that in the figure below.

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 25^(∘)+y^(∘)=90^(∘).Let's solve this equation.

25^(∘)+y^(∘)=90^(∘)

SubEqn

LHS-25^(∘)=RHS-25^(∘)

y^(∘)=65^(∘)

To find the measure of x^(∘), we will have to label more angles in the diagram. We will call these angles w^(∘) and z^(∘).

Note that w^(∘) and z^(∘) are vertical angles, and according to the Vertical Angles Congruence Theorem, vertical angles are congruent. Therefore, we have z^(∘)=w^(∘). To find the value of z^(∘), we can use the Triangle Sum Theorem which says that the sum of the measures of the interior angles of a triangle is 180^(∘). z^(∘)+90^(∘)+20^(∘)=180^(∘). Let's solve this equation.

z^(∘)+90^(∘)+20^(∘)=180^(∘)

AddTerms

Add terms

z^(∘)+110^(∘)=180^(∘)

SubEqn

LHS-110^(∘)=RHS-110^(∘)

z^(∘)=70^(∘)

As we already discussed, w^(∘) and z^(∘) are vertical angles which means we have z^(∘)=70^(∘). Let's complete our figure with this information.

Now we can use the Triangle Sum Theorem again to find the value of x^(∘). We get the following equation x^(∘)+70^(∘)+25^(∘)=180^(∘). Let's solve this equation.

x^(∘)+70^(∘)+25^(∘)=180^(∘)

AddTerms

Add terms

x^(∘)+95^(∘)=180^(∘)

SubEqn

LHS-95^(∘)=RHS-95^(∘)

x^(∘)=85^(∘)

The measure of x^(∘) is 85^(∘)