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In quadrilateral WXYZ, [tex]WC = 2x + 5[/tex] and [tex]CY = 3x + 2[/tex].

What must [tex]x[/tex] equal for quadrilateral WXYZ to be a parallelogram?

[tex]x =[/tex]

Answer :

To determine the value of [tex]$x$[/tex] for which quadrilateral [tex]$WXYZ$[/tex] is a parallelogram, we use the property that in a parallelogram the diagonals bisect each other. This means that the segments formed by the intersection of the diagonals are congruent.

Since points [tex]$W$[/tex], [tex]$C$[/tex], and [tex]$Y$[/tex] are along one diagonal, we have:

[tex]$$
WC = CY
$$[/tex]

Given that:

[tex]$$
WC = 2x + 5 \quad \text{and} \quad CY = 3x + 2,
$$[/tex]

we set them equal to each other:

[tex]$$
2x + 5 = 3x + 2.
$$[/tex]

Now, solve for [tex]$x$[/tex] step-by-step:

1. Subtract [tex]$2x$[/tex] from both sides:

[tex]$$
5 = x + 2.
$$[/tex]

2. Subtract [tex]$2$[/tex] from both sides:

[tex]$$
x = 3.
$$[/tex]

To verify, substitute [tex]$x = 3$[/tex] into the expressions for [tex]$WC$[/tex] and [tex]$CY$[/tex]:

- For [tex]$WC$[/tex]:

[tex]$$
2(3) + 5 = 6 + 5 = 11,
$$[/tex]

- For [tex]$CY$[/tex]:

[tex]$$
3(3) + 2 = 9 + 2 = 11.
$$[/tex]

Both segments are equal, confirming that [tex]$x = 3$[/tex] is the correct solution.

Hence, the value of [tex]$x$[/tex] must be [tex]$3$[/tex] for the quadrilateral [tex]$WXYZ$[/tex] to be a parallelogram.

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