The new image appears to show a similar geometric problem with a circle and an inscribed triangle. Let’s analyze this one as well.

Problem Description

• There is a circle with center $O$.
• Triangle $ABC$ is inscribed in the circle.
• The angle at $C$ is given as $70^\circ$.
• The goal is to find the measure of $\angle AOB$.

Step-by-Step Solution

1. Identify the Given Angle:

   • $\angle ACB = 70^\circ$.

2. Use the Inscribed Angle Theorem:

   • The Inscribed Angle Theorem states that an angle inscribed in a circle is half the measure of the central angle that subtends the same arc.
   • Thus, the central angle $\angle AOB$ is twice the inscribed angle $\angle ACB$.

3. Calculate the Central Angle:

   • $\angle AOB = 2 \times \angle ACB$
   • $\angle AOB = 2 \times 70^\circ = 140^\circ$

So, the measure of $\angle AOB$ is $140^\circ$.

Additional Explanation:

• The Inscribed Angle Theorem is crucial here. It helps us relate the inscribed angle (an angle with its vertex on the circle) to the central angle (an angle with its vertex at the center of the circle).
• Knowing the measure of $\angle ACB$ (an inscribed angle), we can directly find the measure of the corresponding central angle $\angle AOB$ by doubling it.

Do you have any more specific questions about this problem, or is there any more context available from the question?

Follow-up Questions:

1. What is the Inscribed Angle Theorem?
2. How does the central angle relate to the inscribed angle in a circle?
3. Can you find other angles in the triangle using the information given?
4. What is the significance of an angle inscribed in a semicircle?
5. How can the properties of a cyclic quadrilateral help solve this problem?
6. What other problems can be solved using the Inscribed Angle Theorem?
7. How do the properties of chords and arcs relate to inscribed angles?
8. How can symmetry in geometric figures simplify solving problems?

Tip:

When solving geometric problems involving circles, always look for ways to apply the Inscribed Angle Theorem and other circle theorems. These tools can provide quick insights and simplify the problem-solving process.