Solve for x and y given the parallelogram.

Steps to Solve

1. Identify angle relationships in the parallelogram

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (add up to 180°). Here, we can establish two equations based on the given angles.

2. Set up the first equation using consecutive angles

The angle represented as $(2x + 30)^\circ$ and the angle represented as $(5x - 9)^\circ$ are consecutive angles.

The relationship can be expressed as:
$$(2x + 30) + (5x - 9) = 180$$

3. Simplify and solve for x

Combine like terms in the equation and isolate (x):

$$2x + 30 + 5x - 9 = 180$$
$$7x + 21 = 180$$
$$7x = 180 - 21$$
$$7x = 159$$
$$x = \frac{159}{7}$$
$$x = 22.71$$ (approximately)

4. Set up the second equation using the other angles

The angle represented as $\left(\frac{1}{2}y\right)^\circ$ and the angle represented as $(y - 12)^\circ$ are also consecutive angles.

The relationship can be expressed as:
$$\frac{1}{2}y + (y - 12) = 180$$

5. Simplify and solve for y

Combine like terms in the equation and isolate (y):

$$\frac{1}{2}y + y - 12 = 180$$
$$\frac{3}{2}y - 12 = 180$$
$$\frac{3}{2}y = 180 + 12$$
$$\frac{3}{2}y = 192$$
$$y = \frac{192 \cdot 2}{3}$$
$$y = \frac{384}{3}$$
$$y = 128$$