We are given a quadrilateral $ABCD$ with side lengths $AD = 4y - 20$, $AB = 8x - 34$, $BC = 6x - 4$, and $CD = 9x - 40$. We are asked to find the values of $x$ and $y$ that make $ABCD$ a parallelogram.

1. Problem Description

We are given a quadrilateral $ABCD$ with side lengths $AD = 4y - 20$, $AB = 8x - 34$, $BC = 6x - 4$, and $CD = 9x - 40$. We are asked to find the values of $x$ and $y$ that make $ABCD$ a parallelogram.

2. Solution Steps

In a parallelogram, opposite sides are equal in length. Therefore, we have:

$AD = BC$ and $AB = CD$.

Substituting the given expressions for the side lengths, we get the following two equations:

$4y - 20 = 6x - 4$

$8x - 34 = 9x - 40$

First, we solve the second equation for $x$:

$8x - 34 = 9x - 40$

$40 - 34 = 9x - 8x$

$6 = x$

Thus, $x = 6$.

Next, we substitute $x = 6$ into the first equation to solve for $y$:

$4y - 20 = 6(6) - 4$

$4y - 20 = 36 - 4$

$4y - 20 = 32$

$4y = 32 + 20$

$4y = 52$

$y = \frac{52}{4}$

$y = 13$

Thus, $y = 13$.

3. Final Answer

$x = 6$

$y = 13$