Basic Rules of angles

The top and bottom lines are parallel, indicated by the double hash marks.
The angle $8x$ is vertically opposite to the interior angle at the bottom-left vertex. Therefore, the interior angle at the bottom-left is $8x$.
The angle $\omega$ is vertically opposite to the angle formed by the left transversal and the top parallel line. This angle is composed of $x$ and $y$, so the interior angle at the top-left vertex is $x+y$. Thus, $\omega = x+y$.
The angle at the bottom-right vertex is $90^\circ$, indicated by the square symbol.
The angle $\beta$ is vertically opposite to the interior angle at the top-right vertex. Let's call the interior angle at the top-right $Z$. So, $\beta = Z$.

Since the top and bottom lines are parallel, the consecutive interior angles on the right transversal are supplementary.

Since $\beta = Z$, we have:

The consecutive interior angles on the left transversal are supplementary.

The single hash marks indicate that the left slanted side and the diagonal line are equal in length. Let the top-left vertex be A, the bottom-left vertex be B, and the bottom-right vertex be C. The equal sides are AB and AC, forming an isosceles triangle ABC.
In an isosceles triangle, the angles opposite the equal sides are equal. So, $\angle ABC = \angle ACB$.
The angle $\angle ABC$ is the interior angle at the bottom-left vertex, which is $8x$.
Therefore, $\angle ACB = 8x$. This is the angle formed by the diagonal AC and the bottom parallel line BC.
The angle $\angle BAC$ is the angle labeled $x$ in the diagram.
The sum of angles in triangle ABC is $180^\circ$:

Substitute the value of $x$ into Equation 1:

Now calculate $\omega$ using $\omega = x+y$:

The values of the angles are:

• $x = \boxed{\frac{180}{17}^\circ}$
• $y = \boxed{\frac{1440}{17}^\circ}$
• $\omega = \boxed{\frac{1620}{17}^\circ}$
• $\beta = \boxed{90^\circ}$