We can start this problem by drawing the height and labeling the lengths with the given values.

When we do this, we can see that we have drawn a triangle inside the paralellogram including \(\displaystyle \measuredangle A\). Because we know the lengths of two sides of this triangle, we can use trigonometry to find \(\displaystyle \measuredangle A\).

With respect to \(\displaystyle \measuredangle A\), we know the values of the opposite and hypotenuse sides of the triangle. Thus, we can use the sine function to solve for \(\displaystyle \measuredangle A\).

\(\displaystyle \sin \left (\measuredangle A\right ) = \frac{\textup{opposite}}{\textup{hypotenuse}}\)

\(\displaystyle \sin \left (\measuredangle A\right ) = \frac{12}{15}= 0.8\)

\(\displaystyle \measuredangle A = \arcsin\left ( 0.8\right ) = 53.1^{\circ}\)

In a parallelogram, consecutive angles are supplementary. Thus,

\(\displaystyle \measuredangle B + \measuredangle C = 180^{\circ}\)

\(\displaystyle \left ( 2x + 15\right ) + x = 180\)

\(\displaystyle 3x + 15 = 180\)

\(\displaystyle 3x = 165\)

\(\displaystyle x = 55\)

In a parallelogram, consecutive angles are supplementary (i.e. add to \(\displaystyle 180^{\circ}\)) and opposite angles are congruent (i.e. equal). Using these properties, we can write a system of equations.

1. \(\displaystyle \measuredangle B = \measuredangle D\)

2. \(\displaystyle \measuredangle C + \measuredangle D = 180^{\circ}\)

Starting with equation 1.,

\(\displaystyle 2x + 3 = y + x\)

\(\displaystyle x + 3 = y\)

\(\displaystyle x = y - 3\)

Now substituting into equation 2.,

\(\displaystyle y + \left ( y + x\right ) = 180\)

\(\displaystyle 2y + x = 2y + (y - 3) = 180\)

\(\displaystyle 3y - 3 = 180\)

\(\displaystyle 3y = 183\)

\(\displaystyle y = 61\)

Finally, because opposite angles are congruent, we know that \(\displaystyle \measuredangle A = \measuredangle C\).

\(\displaystyle z = y\)

\(\displaystyle z = 61\)

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent. Using these properties, we can write a system of equations.

1. \(\displaystyle \measuredangle B =\measuredangle D\)

2. \(\displaystyle \measuredangle B + \measuredangle C = 180^{\circ}\)

3. \(\displaystyle \measuredangle A = \measuredangle C\)

Starting with equation 1.,

\(\displaystyle x = 4y - x\)

\(\displaystyle 2x = 4y\)

\(\displaystyle x = 2y\)

Substituting into equation 2.,

\(\displaystyle x + y = 180\)

\(\displaystyle 2y + y = 180\)

\(\displaystyle 3y = 180\)

\(\displaystyle y = 60\)

Using equation 3.,

\(\displaystyle z = y\)

\(\displaystyle z = 60\)

In a parellelogram, consecutive angles are supplementary.

\(\displaystyle \measuredangle C + \measuredangle D = 180^{\circ}\)

\(\displaystyle 75^{\circ} + \measuredangle D = 180^{\circ}\)

\(\displaystyle \measuredangle D = 105^{\circ}\)

In a parallelogram, opposite angles are congruent.

\(\displaystyle \measuredangle B = \measuredangle D\)

\(\displaystyle \measuredangle B = 125^{\circ}\)

In a parallelogram, consecutive angles are supplementary and opposite angles are congruent.

\(\displaystyle \measuredangle A + \measuredangle D = 180^{\circ}\)

\(\displaystyle z + \left ( 2z + 9\right ) = 180\)

\(\displaystyle 3z + 9 = 180\)

\(\displaystyle 3z = 171\)

\(\displaystyle z = 57\)

\(\displaystyle \measuredangle A = \measuredangle C\)

\(\displaystyle z = y\)

\(\displaystyle 57 = y\)

In the above parallelogram, \(\displaystyle \measuredangle A\) and \(\displaystyle \measuredangle B\) are consecutive angles (i.e. next to each other). In a parallelogram, consecutive angles are supplementary, meaning they add to \(\displaystyle 180^{\circ}\).

\(\displaystyle \measuredangle A + \measuredangle B = 180^{\circ}\)

\(\displaystyle 45^{\circ} + \measuredangle B = 180^{\circ}\)

\(\displaystyle \measuredangle B = 135^{\circ}\)

In parallelogram \(\displaystyle ABCD\), \(\displaystyle \measuredangle B\) and \(\displaystyle \measuredangle D\) are opposite angles. In a parallelogram, opposite angles are congruent. This means these two angles are equal.

\(\displaystyle \measuredangle D = \measuredangle B\)

\(\displaystyle \measuredangle D = 117^{\circ}\)

In a parallelogram, consecutive angles are supplementary.  \(\displaystyle \measuredangle A\) and \(\displaystyle \measuredangle D\) are consecutive, so their sum is \(\displaystyle 180^{\circ}\).