Answer :
To prove that quadrilateral [tex]$WXYZ$[/tex] is a parallelogram, we use the fact that in a parallelogram the diagonals bisect each other. This means that the segments into which one diagonal is divided are equal in length.
You are given the segments:
[tex]$$
WC = 2x+5 \quad \text{and} \quad CY = 3x+2.
$$[/tex]
Since the diagonal is bisected, we have:
[tex]$$
2x+5 = 3x+2.
$$[/tex]
Now, solve this equation step by step:
1. Subtract [tex]$2x$[/tex] from both sides:
[tex]$$
5 = x+2.
$$[/tex]
2. Subtract [tex]$2$[/tex] from both sides to isolate [tex]$x$[/tex]:
[tex]$$
x = 3.
$$[/tex]
Thus, the value of [tex]$x$[/tex] must be [tex]$3$[/tex].
To verify, substitute [tex]$x=3$[/tex] back into both expressions:
[tex]$$
WC = 2(3) + 5 = 6 + 5 = 11,
$$[/tex]
[tex]$$
CY = 3(3) + 2 = 9 + 2 = 11.
$$[/tex]
Since [tex]$WC = CY = 11$[/tex], the diagonals bisect each other, confirming that [tex]$WXYZ$[/tex] is a parallelogram when [tex]$x = 3$[/tex].
You are given the segments:
[tex]$$
WC = 2x+5 \quad \text{and} \quad CY = 3x+2.
$$[/tex]
Since the diagonal is bisected, we have:
[tex]$$
2x+5 = 3x+2.
$$[/tex]
Now, solve this equation step by step:
1. Subtract [tex]$2x$[/tex] from both sides:
[tex]$$
5 = x+2.
$$[/tex]
2. Subtract [tex]$2$[/tex] from both sides to isolate [tex]$x$[/tex]:
[tex]$$
x = 3.
$$[/tex]
Thus, the value of [tex]$x$[/tex] must be [tex]$3$[/tex].
To verify, substitute [tex]$x=3$[/tex] back into both expressions:
[tex]$$
WC = 2(3) + 5 = 6 + 5 = 11,
$$[/tex]
[tex]$$
CY = 3(3) + 2 = 9 + 2 = 11.
$$[/tex]
Since [tex]$WC = CY = 11$[/tex], the diagonals bisect each other, confirming that [tex]$WXYZ$[/tex] is a parallelogram when [tex]$x = 3$[/tex].