Calculate the marked angles.

The diagram shows two parallel lines intersected by two transversals, forming a "zigzag" shape. The angles are $y$, $82^\circ$, and $60^\circ$.

The zigzag rule states that the sum of the angles pointing in one direction (e.g., left) equals the sum of the angles pointing in the opposite direction (e.g., right).
In this diagram:

• The angle $y$ points to the right.
• The angle $82^\circ$ points to the left.
• The angle $60^\circ$ points to the right.

Sum of angles pointing right = Sum of angles pointing left

The value of $y$ is 22.

: Finding $\theta$

The diagram shows two parallel lines intersected by two transversals, forming another "zigzag" shape. The angles are $70^\circ$, $59^\circ$, and $\theta$.

Draw an auxiliary line through the vertex of the $59^\circ$ angle, parallel to the top and bottom parallel lines. Let's call this auxiliary line $L_3$.
The first segment connects the top parallel line ($L_1$) to $L_3$. The angle $70^\circ$ is given. The alternate interior angle to $70^\circ$ (formed by the first segment and $L_3$) is $70^\circ$. This angle is on the right side of the first segment.
The angle $59^\circ$ is the angle between the first and second segments.
The angle formed by the first segment and $L_3$ on the left side is $180^\circ - 70^\circ = 110^\circ$.
The angle formed by the second segment and $L_3$ on the left side is $110^\circ - 59^\circ = 51^\circ$.
Now, consider the second segment as a transversal between $L_3$ and the bottom parallel line ($L_2$).
The angle formed by the second segment and $L_3$ on the left side is $51^\circ$.
The angle $\theta$ is formed by the second segment and $L_2$ on the left side.
These two angles ($51^\circ$ and $\theta$) are alternate interior angles.

Since they are alternate interior angles:

The value of $\theta$ is 51.