In the image, we have a set of angles formed by intersecting lines. We're asked to solve for $x^\circ$, and using an auxiliary line can help clarify the relationships between the angles.

Step-by-step solution:

1. Identify angle relationships:

   • The two angles 62° and $x^\circ$ are vertically opposite angles because they are on opposite sides of the intersection of the two lines. Thus: $x^\circ = 62^\circ$ because vertically opposite angles are always equal.

2. Angle sum at a point: We can also check this by considering that the sum of angles around a straight line is 180°. The angle marked as 144° and the angle at the intersection of the lines (complementary to $x^\circ$) form a linear pair. This means: $x + 144^\circ = 180^\circ$ Solving this gives: $x = 180^\circ - 144^\circ = 36^\circ$ However, this is not relevant for the vertically opposite angle case.

Thus, the correct measure of $x$ is 62° based on the vertical angles theorem.

Do you want further details or have any questions?

Here are 5 related questions for more practice:

1. If another angle on the same diagram measures 118°, what is its vertically opposite angle?
2. How do you identify vertically opposite angles in more complex diagrams?
3. Can you find all angles formed by two intersecting lines if one angle is given?
4. What is the sum of all interior angles around a point where multiple lines meet?
5. If you are given two angles in a triangle, how do you find the third angle?

Tip: Vertically opposite angles are always equal whenever two lines intersect, regardless of the diagram complexity.