Exercise 122 Page 140

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, we can say that they are congruent by the Corresponding Angles Theorem.

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

.LHS /5.=.RHS /5.

x = 34

d Since this is an isosceles triangle we can, by the Base Angles Theorem, say that the base angles are congruent. Therefore, if one is labeled x, the other must also be x.

Using the Triangle Angle Sum Theorem, we know that the sum of a triangle's three angle measures is 180^(∘). With this, we can write an equation. x+x+(2x-6^(∘))=180^(∘). Let's solve for x in this equation.

x+x+(2x-6)=180

RemovePar

Remove parentheses

x+x+2x-6=180

AddTerms

Add terms

4x-6=180

AddEqn

LHS+6=RHS+6

4x=186

DivEqn

.LHS /4.=.RHS /4.