Let's analyze the diagram step-by-step.

Given Information:

1. Lines $a$ and $b$ are parallel.
2. Angles $2$ and $4$ each measure $32^\circ$.
3. We need to determine the value of $x$ so that lines $l$ and $m$ are parallel.

Steps to Solve:

Step 1: Use the properties of parallel lines and corresponding angles.

• For $l$ and $m$ to be parallel, the corresponding angles formed by a transversal must be equal.

Step 2: Analyze the geometry of the setup.

• Angle $x$ is a combination of angle $1$ and angle $3$.
• Angles $2$ and $4$ are given as $32^\circ$.

Step 3: Relationship between angles.

• The sum of the angles on a straight line is $180^\circ$. Using this property: $\text{Angle } 1 = 180^\circ - \text{Angle } 2 = 180^\circ - 32^\circ = 148^\circ.$

Step 4: Confirm the parallelism condition.

• If $l$ and $m$ are parallel, $x = \text{Angle } 1 + \text{Angle } 3$. Since angle $3 = 32^\circ$, we calculate: $x = 148^\circ + 32^\circ = 148^\circ.$

Final Answer:

The value of $x$ is 148.

Would you like a more detailed breakdown or have other questions?
Here are five related questions to consider:

1. How can we generalize this problem for other angle measures?
2. What are alternate methods to verify the parallelism condition?
3. How does this problem change if angle $3$ is unknown?
4. Why do corresponding angles determine parallelism?
5. Can we apply trigonometry to verify these angle relationships?

Tip: Always use properties of supplementary and corresponding angles to check parallelism in geometric diagrams.