Exercise 20 Page 532

Let's find the value of each variable one at a time. We will begin by naming the vertices of the given parallelogram.

Value of x

The sides with lengths 2x + 9 and 4x - 5 are opposite sides in our parallelogram. Recall the Theorem 6.3, which tells us that if a quadrilateral is a parallelogram, then its opposite sides are congruent. Therefore, the lengths of the opposite sides are equal. 2x + 9= 4x - 5 Let's solve above equation to find the value of x.

2x + 9 = 4x - 5

SubEqn

LHS-9=RHS-9

2x = 4x - 14

SubEqn

LHS-4x=RHS-4x

- 2x = - 14

DivEqn

.LHS /(- 2).=.RHS /(- 2).

x = 7

Value of y

We see that ∠ ADB and ∠ DBC are alternate interior angles. Similarly, ∠ BDC and ∠ ABD are also alternate interior angles. Opposite sides of a parallelogram are parallel, so by the Alternate Interior Angles Theorem we can conclude that they are congruent.

Notice that ∠ CAB, ∠ ABC, and ∠ BCA are three angles in a triangle. By the Angle Sum Theorem, we can conclude that their measures add to 180. m ∠ CAB + m ∠ ABC + m∠ BCA = 180 We already know that m ∠ CAB = 4y, m ∠ ABC = 23 + 42, and m∠ BCA= 83. Let's substitute these values into our equation to find y.

m ∠ CAB + m ∠ ABC + m∠ BCA = 180

SubstituteValues

Substitute values

4y + 23 + 42 + 83 = 180

AddTerms

Add terms

4y + 148 = 180

SubEqn

LHS-148=RHS-148

4y = 32

DivEqn

.LHS /4.=.RHS /4.

y = 8