Exercise 119 Page 65

a Examining the diagram, we see two parallel lines cut by a transversal. The labeled angles have corresponding positions relative to the transversal, which means they are corresponding angles. Because the lines are parallel, by the Corresponding Angles Theorem we can say that they are congruent.

Since the angles are congruent, we can equate their measures and solve for x. x+15^(∘)=102^(∘) ⇔ x=87^(∘)

b The pair of labeled angles is an example of vertical angles. By the Vertical Angles Congruence Theorem, we know these angles are congruent.

Since the angles are congruent, we can equate the expressions for their measures. 7x-3^(∘) = x+21^(∘) Let's solve for x in this equation.

7x-3=x+21

AddEqn

LHS+3=RHS+3

7x = x+24

SubEqn

LHS-x=RHS-x

6x = 24

DivEqn

.LHS /6.=.RHS /6.

x = 4

c Examining the diagram, we see that the labeled angles form a straight line, making them supplementary angles. This means that their measures sum to be 180^(∘). With this we can write an equation.

(3x+8^(∘))+(2x+2^(∘)) = 180^(∘) Let's solve this equation for x.

(3x+8)+(2x+2) = 180

RemovePar

Remove parentheses

3x+8+2x+2 = 180

AddTerms

Add terms

5x+10 = 180

SubEqn

LHS-10=RHS-10

5x = 170

DivEqn