The problem shows two parallel lines cut by a transversal, forming several angles. Given values are:

• $12^\circ$
• $20^\circ$
• $40^\circ$
• an unknown angle $x$

To solve for $x$, let's analyze the relationships between the angles.

1. The angle labeled $40^\circ$ and $x$ are corresponding angles because they are on the same side of the transversal and between the two parallel lines. Therefore, they are equal. This gives us: $x = 40^\circ$

Solution:

$x = 40^\circ$

Would you like more details or have any further questions?

Here are some additional related questions to explore:

1. How do corresponding angles help in solving for unknown angles in parallel line configurations?
2. What are alternate interior angles, and how are they used in similar geometry problems?
3. Can you explain the difference between corresponding and co-interior angles?
4. How would you solve for an unknown angle if only two angles were given?
5. What is the sum of angles on a straight line, and how can it help solve for unknown angles?

Tip: Remember that in parallel line problems, corresponding, alternate, and co-interior angles have specific relationships that make finding unknown angles easier.