Exercise 19 Page 500

Let's find the value of each variable one at a time.

Value of x

Notice that the sides with lengths 2x-1 and 4x-17 are opposite sides. Recall the Parallelogram Opposite Sides Theorem says that if a quadrilateral is a parallelogram, then its opposite sides are congruent. Therefore, for the figure to be a parallelogram, the lengths of the opposite sides must be equal. 2x-1=4x-17Let's solve it!

2x-1=4x-17

SubEqn

LHS-2x=RHS-2x

- 1=2x-17

AddEqn

LHS+17=RHS+17

16=2x

DivEqn

.LHS /2.=.RHS /2.

8=x

RearrangeEqn

Rearrange equation

x=8

Value of y

In a parallelogram, the diagonal divides the figure into two congruent triangles. Let's have a look at the given parallelogram. For convenience reasons, we will name the vertices.

For the quadrilateral to be a parallelogram, we need the triangles that make it to be congruent. △ ABD ≅ △ CDB The triangles share the side BD and we have already found the value of x for which AB ≅ CD. As we can see, ∠ ABD and ∠ CDB are included between the congruent sides. If we show that they are congruent, then △ ABD ≅ △ CDB by Side-Angle-Side Congruence Theorem. 5y-13=3y+5 Let's find the value of y for which the angles are congruent.

5y-13=3y+5

SubEqn

LHS-3y=RHS-3y

2y-13=5

AddEqn

LHS+13=RHS+13

2y=18

DivEqn

.LHS /2.=.RHS /2.

y=9