The problem asks to find the value of $x$ such that lines $\ell \parallel m$ (meaning lines $\ell$ and $m$ are parallel).

In the diagram, two angles are given:

• $(7x + 5)^\circ$ on line $\ell$,
• $(5x + 19)^\circ$ on line $m$.

Since $\ell \parallel m$, and these two angles are corresponding angles (formed by the transversal $k$ crossing $\ell$ and $m$), they must be equal.

Thus, we can set up the equation: $7x + 5 = 5x + 19$

Now, solve for $x$:

1. Subtract $5x$ from both sides: $7x - 5x + 5 = 19$ $2x + 5 = 19$

2. Subtract 5 from both sides: $2x = 14$

3. Divide by 2: $x = 7$

So, the value of $x$ is $\boxed{7}$.

Let me know if you'd like more details or have any other questions!

Here are 5 related questions:

1. What are corresponding angles, and how do they help in determining parallel lines?
2. Can alternate interior angles also help in proving lines are parallel?
3. If the angles were on the same side of the transversal but were supplementary, what would that tell us about the lines?
4. How do we apply angle properties to find unknown variables in geometric figures?
5. What are other angle relationships when two parallel lines are crossed by a transversal?

Tip: Always look for angle relationships like corresponding, alternate interior, or same-side interior when dealing with parallel lines and transversals.