In the image, we see a geometric figure consisting of triangles with various labeled angles and points. Let's break it down based on the given information:

1. Triangle $\triangle ABC$:

   • Angle $\angle A = 80^\circ$
   • Angle $\angle B = 60^\circ$
   • Angle $\angle C$ is not explicitly labeled, but we can calculate it because the sum of angles in any triangle is $180^\circ$.

   $\angle C = 180^\circ - (80^\circ + 60^\circ) = 40^\circ$

2. Triangle within triangle $\triangle ABC$:

   • There is a smaller triangle with angles labeled:
     • One angle $86^\circ$
     • Another angle $40^\circ$

3. An unknown angle $x$ is also labeled at the base near point $C$.

Now, since the sum of angles in any triangle is always $180^\circ$, we can calculate the missing angles in the smaller triangle if needed. Let's determine the value of the unknown angle $x$:

In the smaller triangle that includes the angles $40^\circ$ and $86^\circ$:

$\text{Sum of angles} = 180^\circ$

Thus, the remaining angle, which is the angle $x$, can be calculated as:

$x = 180^\circ - (40^\circ + 86^\circ) = 54^\circ$

Summary:

• Angle $x = 54^\circ$ based on the smaller triangle's angle sum.
• The overall structure and the relationships between the triangles are based on the angle sum property.

Do you have any specific questions or further details you'd like to explore regarding this problem?

Related Questions:

1. How do we calculate the missing side lengths in these triangles?
2. Can we explore the properties of similar triangles from this figure?
3. How would you solve for any other unknown angles in this construction?
4. How can the Law of Sines or Law of Cosines be applied to this triangle?
5. What are the applications of angle bisectors in these types of problems?

Tip:

Always use the triangle angle sum property $180^\circ$ as a starting point when solving for unknown angles in geometric figures.