This image depicts a triangle inscribed in a circle. The two angles shown in the diagram are $30^\circ$ and $20^\circ$, with $O$ being the center of the circle, and point $A$ marked along the circumference.

It looks like you might be working with the properties of a circle and an inscribed triangle, specifically dealing with angles and possibly using the Inscribed Angle Theorem.

Method 1: Use of the Inscribed Angle Theorem

1. Inscribed Angle Theorem states that an inscribed angle is half the measure of the central angle subtended by the same arc. For example:

   • If an angle is subtended by arc $BC$ at the center $O$, the angle formed at the circumference will be half of the central angle.

2. Using the angles shown:

   • Angle $\angle OAC = 30^\circ$ is a central angle.
   • The corresponding inscribed angle subtended by the same arc is $\angle AOC$, which will be $30^\circ \times 2 = 60^\circ$.

3. Calculation of unknown angles:

   • Using the fact that the sum of angles in any triangle is $180^\circ$, if you have two known angles in the triangle, you can subtract from $180^\circ$ to find the third one.
   • For the triangle, if two angles are given, the third angle can be calculated by subtracting the sum of the known angles from $180^\circ$.

Method 2: Direct Use of Central and Inscribed Angle Relationships

The relationship between the central and inscribed angles can help in determining the unknown angles. The central angle formed by the lines from the center $O$ to points on the circle determines the inscribed angles, which can be easily worked with.

Would you like a more detailed breakdown or calculation for this setup? Feel free to ask for any clarifications or further steps!

Follow-up Questions:

1. How do you determine the angle sum in a circle for inscribed angles?
2. Can you explain how to calculate the third angle in the triangle using the Inscribed Angle Theorem?
3. How would you apply the same theorem to other types of geometric figures inscribed in circles?
4. What are the key properties of the angles in an isosceles triangle inscribed in a circle?
5. How do you prove that the sum of angles in any triangle equals 180°?

Tip: When working with inscribed angles, always remember that the angle at the center is twice the inscribed angle that subtends the same arc. This principle simplifies many problems!