The image depicts a triangle with a base that appears to be isosceles due to the equal markings on both sides of the base. Several angles are marked within and around the triangle, including:

• A $40^\circ$ angle on the left base side,
• A $20^\circ$ angle on the right base side,
• Two unknown angles $X$ and $Y$ inside the triangle,
• Two more labeled angles $A$ and $B$ around a black circle (which could represent an incenter or another type of point).

Let's break down how to find the unknown angles, particularly $X$, $Y$, $A$, and $B$.

Step-by-Step Solution:

1. Symmetry of the Triangle: Since the base has equal lengths on both sides, it is likely that the triangle is isosceles. This symmetry can help in understanding the relationships between the angles.

2. Angle Sum Property: In any triangle, the sum of the interior angles is always $180^\circ$.

3. Calculating Angle $X$: $\text{Sum of angles in triangle} = 180^\circ$ Let's assume $X + X + 40^\circ = 180^\circ$ for the left triangle formed by the angle. From this, you can find $X$.

4. Use of External Angles $A$ and $B$: External angles might help with $Y$, depending on how the circle is interacting with the triangle.

Would you like to proceed with specific calculations for these angles or a different approach?