Question

based on the measures shown, could the figure be a parallelogram? yes, one pair of opposite sides could measure 10 in., and the other pair could measure 13 in. yes, one pair of opposite sides could measure 10 in., and the other pair could measure 8 in. no, there are three different values for x when each expression is set equal to 10. no, the value of x that makes one pair of sides congruent does not make the other pair of sides congruent.

Explanation:

Step1: Set up equations for opposite - side congruence

In a parallelogram, opposite sides are congruent. So we set up two equations:
Equation 1: \(x + 5=2x - 3\) and Equation 2: \(10=x + 2\).
For Equation 1:
\[

$$\begin{align*} x+5&=2x - 3\\ x-2x&=-3 - 5\\ -x&=-8\\ x&=8 \end{align*}$$

\]
For Equation 2: \(x=10 - 2=8\).
Now, check the other pair of sides.
If \(x = 8\), then \(x + 5=8 + 5=13\) and \(2x-3=2\times8 - 3=16 - 3 = 13\), and \(x + 2=8+2 = 10\).
But if we set \(x + 5 = 10\), then \(x=5\), and \(2x-3=2\times5-3 = 7\) and \(x + 2=5 + 2=7\). The value of \(x\) that makes one pair of sides congruent does not make the other pair of sides congruent.

Answer:

D. No, the value of \(x\) that makes one pair of sides congruent does not make the other pair of sides congruent.