In the image, you are asked to find the value of $x$, with the following information:

• Lines $AB \parallel CD$
• Angle $\angle ABO = 50^\circ$
• Angle $\angle ODC = 30^\circ$

Solution:

Since $AB \parallel CD$, and the line segment $OB$ crosses them, the angles formed on the alternate sides of the transversal are alternate interior angles. This means that $\angle ABO$ and $\angle BOD$ are equal as they are alternate interior angles.

Thus, $\angle BOD = 50^\circ$.

Now, we can use the fact that the sum of the angles in the triangle $OBD$ must equal $180^\circ$. That gives us:

$x + 50^\circ + 30^\circ = 180^\circ$

Simplifying:

$x + 80^\circ = 180^\circ$

$x = 180^\circ - 80^\circ = 100^\circ$

So, the value of $x$ is $\boxed{100^\circ}$.

Would you like further details or clarifications?

Here are some related questions you might want to explore:

1. What are alternate interior angles?
2. How do parallel lines affect angle relationships?
3. Can this problem be solved using any other method?
4. What are the properties of angles in a triangle?
5. How can you verify that the two lines are truly parallel?

Tip: Always check for angle relationships like alternate interior angles when dealing with parallel lines crossed by a transversal.