In this problem, we are dealing with a circle where points $A$, $B$, $C$, and $D$ lie on the circumference, and $AC$ passes through the center $O$, meaning $AC$ is a diameter of the circle. The goal is to find the angles $\angle ACB$ and $\angle ADB$.

1. Angle $\angle ACB$:

Since $AC$ is a diameter, by the Thales' Theorem, the angle subtended by the diameter on the circumference of a circle is always a right angle. This means: $\angle ACB = 90^\circ$

2. Angle $\angle ADB$:

This angle is subtended by the same arc $AB$ as the angle $\angle ACB$, but it is not subtended by the diameter. According to the angle at the center theorem, the angle subtended by an arc at the circumference is half the angle subtended by the same arc at the center.

We know the angle at $A$, $\angle AOB$, at the center is a straight line, so: $\angle AOB = 180^\circ$ Therefore, the angle at the circumference, $\angle ADB$, will be: $\angle ADB = \frac{1}{2} \times \angle AOB = \frac{1}{2} \times 180^\circ = 90^\circ$

Summary of the Angles:

• $\angle ACB = 90^\circ$
• $\angle ADB = 90^\circ$

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

Related Questions:

1. What is Thales' Theorem and how is it applied to circles?
2. Why is the angle subtended by a diameter always $90^\circ$?
3. How does the Angle at the Center Theorem work in various circle problems?
4. Can you calculate the angles for any other points on the circle using similar rules?
5. How would the calculations change if $AC$ were not a diameter?

Tip:

Always look for diameters in circle problems—they frequently lead to right angles, making calculations easier!