Exercise 54 Page 511

Recall the theorem that says that a quadrilateral is a parallelogram if a pair of sides is both parallel and congruent.

x=2
y=41

Practice makes perfect

We want to find the values of x and y so that the quadrilateral is a parallelogram.

Value of x

Recall the theorem.

If a pair of sides of a quadrilateral is both parallel and congruent, then it is a parallelogram.

Therefore, we want the sides measuring 2x+7 and x+9 to have equal length. 2x+7= x+9 Let's solve this equation for x.

2x+7=x+9

SubEqn

LHS-7=RHS-7

2x=x+2

SubEqn

LHS-x=RHS-x

x=2

Value of y

We also want these same sides to be parallel. Recall the theorem.

Converse Consecutive Interior Angles Theorem
If a pair of consecutive consecutive interior angles formed by a transversal are supplementary, then the crossed lines are parallel.

Therefore, we want the angles measuring (2y-5)^(∘) and (2y+21)^(∘) to be supplementary, meaning they add up to 180^(∘). (2y-5)+ (2y+21)=180 Let's solve for y.

(2y-5)+(2y+21)=180

RemovePar

Remove parentheses

2y-5+2y+21=180

AddTerms

Add terms

4y+16=180

SubEqn

LHS-16=RHS-16

4y=164

DivEqn

.LHS /4.=.RHS /4.