Geometry worksheet on angles and parallel lines with diagrams and problems.

To solve this problem, we need to find the value of $x$ based on the given information about supplementary angles.

Step 1: Understand the Problem

We are told that:
- $\angle 1$ and $\angle 2$ are supplementary angles.
- The measure of $\angle 1$ is $4x$.
- The measure of $\angle 2$ is $12x$.

Supplementary angles are two angles whose measures add up to $180^\circ$.

Step 2: Set Up the Equation

Since $\angle 1$ and $\angle 2$ are supplementary, we can write the equation:
$$
\text{Measure of } \angle 1 + \text{Measure of } \angle 2 = 180^\circ
$$

Substituting the given expressions for the angles:
$$
4x + 12x = 180
$$

Step 3: Solve for $x$

Combine like terms on the left side of the equation:
$$
16x = 180
$$

Now, divide both sides by 16 to isolate $x$:
$$
x = \frac{180}{16}
$$

Simplify the fraction:
$$
x = 11.25
$$

Step 4: Verify the Solution

Let's check if the solution is correct by substituting $x = 11.25$ back into the expressions for the angles:
- Measure of $\angle 1 = 4x = 4(11.25) = 45^\circ$
- Measure of $\angle 2 = 12x = 12(11.25) = 135^\circ$

Now, check if the angles are supplementary:
$$
45^\circ + 135^\circ = 180^\circ
$$

The angles are indeed supplementary, so the solution is correct.

Final Answer:

$$
\boxed{11.25}
$$