The image shows a triangle with three labeled angles. The triangle's vertices are labeled as $A$, $B$, and $C$, and the given angle measures are:

• $\angle A = 55^\circ$
• $\angle B = 45^\circ$
• $\angle C = 30^\circ$

You are asked to find the unknown angle $x$, which is also part of the triangle and located near $\angle C$.

Step-by-step solution:

The sum of the interior angles in any triangle is always $180^\circ$. Therefore, the equation for the sum of angles in triangle $ABC$ is:

$\angle A + \angle B + \angle C = 180^\circ$

Substituting the known values:

$55^\circ + 45^\circ + 30^\circ + x = 180^\circ$

Simplifying:

$130^\circ + x = 180^\circ$

Now, solving for $x$:

$x = 180^\circ - 130^\circ = 50^\circ$

Thus, the value of $x$ is:

$x = 50^\circ$

Would you like further details or have any questions?

Here are five related questions to explore this topic:

1. How do you determine angles in a non-right triangle?
2. Can you apply this method to triangles with angles in radians?
3. How would the solution change if one of the angles was obtuse?
4. What are some common mistakes when finding unknown angles in a triangle?
5. How does the triangle inequality theorem relate to angle measurements?

Tip: Always double-check the sum of the angles when solving for an unknown angle in a triangle.