What values must x and y have to make the quadrilateral a parallelogram?

The given diagram is:

In a parallelogram, both pairs of the opposite sides of the parallelogram are parallel.

Therefore, to make the given quadrilateral a parallelogram, both pairs of the opposite sides of the parallelogram are parallel.

The angles measuring $\left(9y\right)°$and$\left(4x+1\right)°$ are the alternate interior angles made by the diagonal which is also the transversal for the parallel sides of the parallelogram.

As the alternate interior angles are congruent.

Therefore, $9y=4x+1$.

The angles measuring $\left(3x\right)°$and$\left(7y-2\right)°$ are the alternate interior angles made by the diagonal which is also the transversal for the parallel sides of the parallelogram.

As the alternate interior angles are congruent.

Therefore, $3x=7y-2$.

Therefore, it can be noticed that:

$\begin{array}{c}9y=4x+1\left(1\right)\\ 3x=7y-2\left(2\right)\end{array}$

Therefore, from the equation (1), it can be obtained as:

$\begin{array}{c}9y=4x+1\\ y=\frac{4x+1}{9}\end{array}$

Now, substitute the value of y in equation (2) to find the value of x.

Therefore, the value of x can be obtained as:

$\begin{array}{c}3x=7\left(\frac{4x+1}{9}\right)-2\\ 3x=\frac{28x+7}{9}-2\\ 3x=\frac{28x+7-18}{9}\\ 27x=28x-11\\ 11=28x-27x\\ 11=x\end{array}$

Therefore, the value of x is 11.

Now, substitute 11 for x in the equation $y=\frac{4x+1}{9}$ to find the value of y.

Therefore, the value of y can be obtained as:

$\begin{array}{c}y=\frac{4\left(11\right)+1}{9}\\ =\frac{44+1}{9}\\ =\frac{45}{9}\\ =5\end{array}$

Therefore, the value of y is 5.