The image you've provided shows a geometric diagram involving a triangle, a circle inscribed within it, and some marked tangents. To determine the angles based on the tangents and the inscribed circle, I will guide you through the steps for analyzing this configuration.

Problem Breakdown:

1. Inscribed Circle (Incircle): The circle inside the triangle is tangent to each side of the triangle. The tangents to the triangle from the same external point are equal in length.
2. Tangents and Triangle Angles: The relationship between the tangents and angles of the triangle can be analyzed using properties of the circle and the triangle's vertices.

Steps for Finding the Angles:

1. Use the fact that the sum of angles in a triangle is always $180^\circ$.
2. Tangents to the incircle: For each vertex, the angle formed by the tangents from the vertex to the points of tangency can help determine the individual triangle angles.
3. Angle Properties: If a tangent line forms an angle with the side of a triangle, then the tangent's angle is supplementary to the angle formed at the point of tangency.

Given that your diagram involves tangents and a triangle, it’s possible to use known geometric rules like the angle bisector theorem and tangent properties to derive the angles. However, for precise calculations, we would need to measure or have more information on specific side lengths or tangent lengths to the circle.

General Formula for Angles:

The angle between two tangents from a point $P$ to a circle is given by: $\theta = \frac{1}{2}( \text{arc subtended by the tangents at the center})$

Would you like to walk through specific geometric constructions from the diagram, or should I provide more examples and details on finding these angles using tangents and geometry? If the goal is to find precise angle measurements, I might need some clarification or additional information on the lengths or specific known angles.

Example questions you might consider after:

1. How can we use tangent properties to find angles in any triangle?
2. What is the relationship between the radius of an incircle and the triangle angles?
3. How does the angle bisector theorem apply to this configuration?
4. Can we generalize the angle measurement technique for other shapes inscribed in circles?
5. How do supplementary angles formed by tangents work in different configurations?

Tip: When working with circles and tangents in a triangle, remember that each tangent touches the circle at exactly one point and forms right angles with the radius at that point of tangency!