Characteristics and Proofs of a Parallelogram: In order for the quadrilateral to be a parallelogram what should the value of the unknown be ?

In a parallelogram, consecutive angles are supplementary, which means they add up to $180^\circ$. Given that $\angle C = 120^\circ$, the angle $\angle D = x$ must fulfill the equation:

$180^\circ = \angle C + \angle D = 120^\circ + x$

Solving for $x$:

$x = 180^\circ - 120^\circ = 60^\circ$

Therefore, the value of $x$ that makes $ABCD$ a parallelogram is $x = 60^\circ$.

The problem requires determining the value of $x$ for the quadrilateral ABCD to be a parallelogram, which involves setting opposite sides of the quadrilateral equal to each other.

For a quadrilateral to be a parallelogram, its opposite sides must be equal. Therefore, we can set the following equality:

$BC = AD$

In the given diagram, it states that $BC = 18$ and $AD = x$. Since opposite sides of a parallelogram are equal, we have:

$x = 18$

Thus, by equating the lengths of the opposite sides, we derive the value of $x$ as $\textbf{18}$.

Therefore, the value of $x$ for quadrilateral ABCD to be a parallelogram is $x = 18$.

To find the values of $x$ and $y$ for the parallelogram ABCD, we use the property that opposite sides of a parallelogram are equal.

1. Opposite sides AB and DC must be equal, therefore:

$x = 15$

2. Opposite sides AD and BC must also be equal, therefore:

$y = 7$

Thus, the values of the unknowns that make ABCD a parallelogram are $x = 15$ and $y = 7$.

This corresponds to choice 1: $x = 15, y = 7$.