In this figure, lines $a$ and $b$ are parallel, and line $t$ is a transversal. The angles labeled $(4x + 36)^\circ$ and $(2x + 18)^\circ$ are corresponding angles, and since line $a \parallel b$, corresponding angles are equal.

Thus, we can set the angles equal to each other:

$4x + 36 = 2x + 18$

Now, let's solve for $x$:

1. Subtract $2x$ from both sides:

$4x - 2x + 36 = 18$

$2x + 36 = 18$

2. Subtract 36 from both sides:

$2x = 18 - 36$

$2x = -18$

3. Divide both sides by 2:

$x = \frac{-18}{2} = -9$

So, the value of $x$ is $-9$.

Would you like further details, or do you have any questions?

Here are 5 related questions to consider:

1. How do alternate interior angles relate when lines are parallel?
2. What is the sum of the interior angles formed by a transversal crossing parallel lines?
3. How would the result change if the angles were supplementary rather than equal?
4. Can the value of $x$ be positive in this type of geometry problem?
5. What are the properties of corresponding angles in parallel line transversals?

Tip: When dealing with parallel lines and transversals, always check the angle relationships—corresponding, alternate interior, or consecutive.