To solve for $x$ in the given diagram, we need to analyze the angles.

Step 1: Use the property of angles on a straight line

The sum of angles on a straight line is always 180°. In the diagram, the angles on the straight line are:

• $108^\circ$ on the left
• An unknown angle adjacent to $x^\circ$
• $30^\circ$ on the right

Thus, the unknown angle adjacent to $x$ (which I'll call $\alpha$) can be found as: $\alpha = 180^\circ - (108^\circ + 30^\circ) = 180^\circ - 138^\circ = 42^\circ$

Step 2: Apply the triangle sum property

In a triangle, the sum of all interior angles is always 180°. In the triangle formed by the angles $x^\circ$, $40^\circ$, and $\alpha$, we know that: $x^\circ + 40^\circ + 42^\circ = 180^\circ$

Solving for $x$: $x = 180^\circ - (40^\circ + 42^\circ) = 180^\circ - 82^\circ = 98^\circ$

Thus, $x = 98^\circ$.

Would you like further details or have any questions?

Here are five related questions:

1. What is the sum of interior angles in a triangle?
2. How do we calculate the exterior angle of a triangle?
3. Can you explain how angles on a straight line sum up to 180°?
4. What is the role of alternate interior angles in parallel lines?
5. How can we calculate the missing angle in a right triangle?

Tip: When working with geometric figures, always look for supplementary angles on a straight line and the sum of angles within a triangle.