Exercise 52 Page 467

We want to find the values of x and y for which RSTV is a parallelogram. To do so, we will use the given algebraic expressions for the segment lengths.

The Parallelogram Diagonals Theorem states that if a quadrilateral is a parallelogram, then its diagonals bisect each other. Therefore, the following segments are congruent. RP ≅ PT and VP ≅ PS By the definition of congruent segments, we can conclude that their lengths are equal. RP = PT and VP = PS Let's create a system of equations by substituting the lengths of the segments into these equations. 2x=y+2 y=x+3 To solve it we will use the Substitution Method.

2x=y+2 & (I) y=x+3 & (II)

Substitute

(I): y= x+3

2x= x+3+2 y=x+3

AddTerms

(I): Add terms

2x=x+5 y=x+3

SubEqn

(I): LHS-x=RHS-x

x=5 y=x+3

Now that we have found x, we can substitute it in Equation (II) to find y.

x=5 & (I) y=x+3 & (II)

Substitute

(II): x= 5

x=5 y= 5+3

AddTerms

(II): Add terms

x=5 y=8