Exercise 14 Page 647

Examining the diagram, we can spot an equiangular triangle and an isosceles triangle. An equiangular triangle has three congruent angles which means each angle has a measure of 180^(∘)/3=60^(∘). Also, according to the Converse of the Base Angles Theorem, if two angles of a triangle are congruent, then the sides opposite them are congruent. Let's add this information to the diagram.

Determining x

The vertex angle of the isosceles triangle forms a linear pair with one of the angles in the equiangular triangle. By the Linear Pair Postulate, we know these angles are supplementary which means they add up to 180^(∘).

8x^(∘)+60^(∘)=180^(∘)

SubEqn

LHS-60^(∘)=RHS-60^(∘)

8x^(∘)=120^(∘)

DivEqn

.LHS /8.=.RHS /8.

x^(∘)=15^(∘)

Determining y

To solve for y, we recognize that one side in the equiangular triangle is 26. According to the Corollary to the Converse of the Base Angles Theorem, if a triangle is equiangular, then it is equilateral.

Since one of the equilateral triangle's side is 26 and another is 5y+1, we can equate these sides and solve for y.

5y+1=26

SubEqn

LHS-1=RHS-1

5y=25

DivEqn

.LHS /5.=.RHS /5.

y=5