To find $x$ in the given triangle, let's analyze the figure and apply angle properties:

1. Parallel Lines and Transversal: The lines $l$ and $m$ are parallel, and the line containing angles $x^\circ$ and $151^\circ$ is a transversal line. Since $l \parallel m$, we know that the interior angles on the same side of the transversal add up to $180^\circ$.

2. Supplementary Angles: Since $x^\circ$ and $151^\circ$ are on the same side of the transversal, they are supplementary: $x + 151^\circ = 180^\circ$

3. Solve for $x$: $x = 180^\circ - 151^\circ = 29^\circ$

Thus, $x = 29^\circ$.

Would you like a further breakdown of the reasoning behind this solution?

Here are some related questions for deeper understanding:

1. What are alternate interior angles, and how are they used with parallel lines?
2. How would the solution change if one of the angles was not given?
3. Can we solve for $x$ using any other properties of triangles?
4. What other types of angle relationships exist with parallel lines and transversals?
5. What is the significance of supplementary angles in geometry?

Tip: Always check for parallel lines and transversals in a diagram, as they often create angle pairs that are equal or supplementary, simplifying the calculations.