The image shows a geometric diagram with a trapezoid-like shape that includes an inner triangle, with the angles and expressions labeled as follows:

• The left angle of the trapezoid is $28x$ degrees.
• The top angle of the inner triangle is $16x$ degrees.
• The bottom right angle of the trapezoid is labeled as $48^\circ$.

Let's solve for $x$ using the information provided:

Step 1: Use the Angle Sum Property

The inner triangle’s angles sum to 180° because it is a triangle. The angles inside the triangle are:

• $16x^\circ$ (top angle of the triangle)
• Two angles adjacent to $l$ and $m$ are supplementary with the angles $28x^\circ$ and $48^\circ$ because they form a straight line.

We can set up the equation by summing the angles in the inner triangle:

$16x + (28x) + (48^\circ) = 180^\circ$

Step 2: Simplify the Equation

Combine like terms:

$16x + 28x + 48 = 180$

$44x + 48 = 180$

Step 3: Solve for $x$

Subtract 48 from both sides:

$44x = 180 - 48$

$44x = 132$

Divide by 44:

$x = 3$

So, the value of $x$ is $3$.

Would you like further details or have any questions? Here are some related topics that could expand on this problem:

1. How to verify the solution using angle relationships.
2. Exploring the properties of trapezoids and triangles.
3. How to handle similar problems involving supplementary angles.
4. The impact of incorrect angle measures on geometric figures.
5. How to identify corresponding and alternate interior angles in complex shapes.

Tip: Always double-check angle relationships, especially when working with complex shapes like trapezoids and triangles, to ensure you account for all properties correctly.