The image shows a triangle split into two smaller triangles, where certain angles are marked. To find the value of $x$, let's analyze the situation:

• The bottom two sides of the triangle are marked as equal, indicating that the triangle is isosceles.
• One base angle of the isosceles triangle is given as $20^\circ$.
• The angle opposite the equal sides is marked as $x$.
• The larger triangle includes a known angle of $40^\circ$ at the bottom left corner.

Steps:

1. Identify the angles of the isosceles triangle: Since the isosceles triangle has two equal angles, and one is given as $20^\circ$, the other base angle is also $20^\circ$.

2. Calculate the third angle of the isosceles triangle (angle at $x$): Using the sum of angles in a triangle, which is always $180^\circ$, we can calculate $x$: $x = 180^\circ - 20^\circ - 20^\circ = 140^\circ$

3. Check the entire triangle: The remaining smaller triangle has an angle of $40^\circ$, and it adds up to complete the angles of the large triangle.

Thus, $x = 140^\circ$.

Let me know if you would like further details or have any questions!

Here are some follow-up questions:

1. What is the general formula for the sum of angles in a triangle?
2. How can you determine if a triangle is isosceles based on side lengths or angles?
3. What are some properties of isosceles triangles regarding their angles?
4. Can you calculate the remaining angle in the large triangle in the diagram?
5. What happens to the angles in other polygons like quadrilaterals or pentagons?

Tip: Always remember that in any triangle, the sum of all three interior angles is $180^\circ$.