Exercise 20 Page 422

We want to find the values of x and y for which ABCD is a parallelogram, using the given algebraic expressions for the segment lengths.

Recall the theorem referenced in the book, which tells us that if a quadrilateral is a parallelogram, then its diagonals bisect each other. Therefore, the following segments are congruent. AT ≅ TC and BT ≅ TD By the definition of congruent segments, we can conclude that their lengths are equal. AT = TC and BT = TD Let's create a system of equations by substituting the lengths of the segments into these equations. 3y-3=3x 4x-2=3y-1 To solve it we will use the Substitution Method.

3y-3=3x & (I) 4x-2=3y-1 & (II)

DivEqn

(I): .LHS /3.=.RHS /3.

y-1=x 4x-2=3y-1

RearrangeEqn

(I): Rearrange equation

x=y-1 4x-2=3y-1

Substitute

(II): x= y-1

x=y-1 4( y-1)-2=3y-1

▼

(II): Solve for y

Distr

(II): Distribute 4

x=y-1 4y-4-2=3y-1

SubTerm

(II): Subtract term

x=y-1 4y-6=3y-1

AddEqn

(II): LHS+6=RHS+6

x=y-1 4y=3y+5

SubEqn

(II): LHS-3y=RHS-3y

x=y-1 y=5

Now that we have found y, we can substitute it in the first equation to find x.

x=y-1 y=5

Substitute

(I): y= 5

x= 5-1 y=5

SubTerm

(I): Subtract term

x=4 y=5