Exercise 50 Page 396

We want to find the values of x and y using on the following diagram.

First, let's focus on the three adjacent angles in the top of the diagram.

These angles sum to a straight angle, so their measures sum to the measure of a straight angle — 180^(∘). Also, notice that the middle adjacent angle is a right angle, so its measure is 90^(∘). We can write the following equation for x using these facts. 7x^(∘) + 90^(∘) + 20^(∘) = 180^(∘) Let's solve it!

7x^(∘) + 90^(∘)+ 20^(∘) = 180^(∘)

AddTerms

Add terms

7x^(∘) + 110^(∘) = 180^(∘)

SubEqn

LHS-110^(∘)=RHS-110^(∘)

7x^(∘) = 70^(∘)

DivEqn

.LHS /7.=.RHS /7.

x^(∘) = 10^(∘)

We have found that x=10. Now, let's find the value of y. To do so, we can notice that the two angles at the bottom of the diagram are adjacent angles. The angle they form is vertical with the right angle at the top of the diagram.

Since vertical angles have equal measure, we can write the following equation using the measures of the two bottom angles. 5x^(∘) + 2y^(∘) = 90^(∘) We already know that x = 10^(∘). Let's substitute 10^(∘) for x in the above equation and solve the resulting equation for y.

5x^(∘) + 2y^(∘) = 90^(∘)

Substitute

x= 10

5( 10)^(∘) + 2y^(∘) = 90^(∘)

Multiply

Multiply

50^(∘) + 2y^(∘) = 90^(∘)

SubEqn

LHS-50^(∘)=RHS-50^(∘)

2y^(∘) = 40^(∘)

DivEqn

.LHS /2.=.RHS /2.

y^(∘)= 20^(∘)

We found that y = 20.