In the given image, two parallel lines $l$ and $m$ are shown, with a transversal intersecting them. The angles $50^\circ$, $30^\circ$, and $x$ are marked. Let's solve for the value of $x$.

Step-by-Step Solution:

1. Identify the relationship between the angles:

   • Since $l \parallel m$, the angles created by the transversal obey the properties of parallel lines and transversals.
   • At point $A$, the $50^\circ$ angle and the angle adjacent to $x$ form a straight line. Hence, they are supplementary.
   • At point $B$, the $30^\circ$ angle is alternate interior to the angle adjacent to $x$, so they are equal.

2. Apply the angle properties:

   • The angle adjacent to $x$ at point $A$ is equal to $180^\circ - 50^\circ = 130^\circ$.
   • Now, the sum of angles in the triangle formed by the transversal and the parallel lines is $180^\circ$. The three angles in the triangle are $x$, $130^\circ$, and $30^\circ$.

3. Solve for $x$: Using the triangle angle sum property: $x + 130^\circ + 30^\circ = 180^\circ$ Simplify: $x + 160^\circ = 180^\circ$ Subtract $160^\circ$ from both sides: $x = 20^\circ$

Final Answer:

$x = 20^\circ$

Would you like a detailed explanation of any step or additional examples? Let me know!

Here are 5 related questions you can explore:

1. What are the properties of angles formed by parallel lines and transversals?
2. How do alternate interior angles relate to parallel lines?
3. What is the sum of the interior angles in a triangle?
4. How do supplementary angles work in geometry?
5. How can you use the properties of parallel lines to solve problems with multiple transversals?

Tip: Always look for supplementary or alternate interior angle relationships when working with parallel lines and transversals.